Why do Buddhists believe that the Buddha was enlightened, despite that he claimed it himself?

I think the most telling story told about the Buddha saying who he was and what his condition was when he responded to the question: "Who are you." The Buddha silently contemplated and then stated "I am awake!" It is unfortunate that we must rely on only oral transmissions (or the texts drawn from oral transmissions) of Shakyamuni Buddha so we cannot definitively quote his words in this story or in other texts with certainty. If this was a true story of the Buddha, it seems that it would be unlikely for him to claim enlightenment for himself. I don't see evidence that he saw himself as the sole Buddha apart from the people but as one of the people realizing his fullest potential. Some say his greatest sermon was holding up the flower. http://kuzanzen.org/2013/02/the-buddha-twirls-a-flower/ Only one student of his knew the significance. This is an example how important silent transmission of the dharma was to Shakyamuni Buddha.

Truth is unless you can get into a time machine and personally encounter the Buddha you can never know for sure. However, The Buddha was a historical figure his teachings were preserved. Read them and judge for yourself if the Buddha was exactly what he claimed to be or not. Even better, actually do the practice and see if it works for you.

Claiming to be enlightened by itself doesn't mean that it's not true. It's just that even during The Buddha's time when He was alive and teaching there weren't that many arahants in the world (maybe a few thousand or so). Let alone now in modern times, long after the pure dhamma has disappeared. An arahant is rare.

The Buddha had mentioned teachers and beings who falsely claimed to be all-knowing and enlightened.

It is possible for someone to achieve higher states but not the ending of mental fermentations and mistakenly believe themselves to be enlightened. It is also possible for non-arahants to achieve iddhi powers and many feats.

If I were to encounter someone who claims to be an arahant or fully enlightened I would ask them this simple question "How do you distinguish arahantship from achieving higher states without achieving arahantship?".

Although I myself can answer the question I wonder how the arahant claimant would respond.

But I believe The Buddha was who he claimed to be, not simply a paccekabuddha but a Sammāsambuddha, meaning both fully enlightened independently and a supreme teacher because:

• No other teacher in all of recorded history ever debated with so many people of different views and explained things in such a fearless, doubtless manner
• No other teacher in all of recorded history ever reasoned things in the ingenious way that The Buddha had
• No other teacher in all of recorded history ever explained things in the detail that The Buddha had
• My personal experiences matched in closely to what The Buddha had said

I've read stuff from other spiritual teachers and those who claim to be enlightened but still they don't compare to Gautama Buddha's teachings. Some may be paccekabuddhas but their teachings are limited and aren't as well-reasoned and thought-out as Gautama Buddha's.

My personal experiences match in closely to what I read from The Buddha. When I concentrate I can accomplish certain things, like healing myself. I believe I've achieved at least something like the first jhana but am not sure if it's exactly what The Buddha described as the first jhana since he describes many different states (see Pancakanga Sutta).

I just know that when I concentrate on destroying painful feelings or fermentations that I feel this strange energy go from my forehead to the top of my head, and if I keep trying to eliminate more fermentations I go higher and higher. What I experience then is a happiness unlike any other.

I feel doubtless, fearless, sorrowless, angerless, full of confidence, calm, unstoppable, full of enjoyment, such an extreme type of enjoyment.

Once I had experienced this type of enjoyment I thought this was the correct path and no other form of happiness, bliss, or enjoyment was worth pursuing.

It is entirely different from the other blissful states I had experienced (like deep relaxation, or the dream-like bliss) which are like blissful states but lacking energy and enjoyment.

Maybe some day I'll find out everything and the whole truth...

Some people may believe so to warring degrees of confidence, but this confidence only becomes unshakable at stream entry. When practicing the Dhamma you start seeing the milestones in the journey your confidence in reinforced and at stream entry it becomes unshakable.

If Buddha has shown the path to Nirvana and when you follow it you get there then you can be sure that the person knows.

The overall idea seems to be that if a being is actually enlightened, then this fact will be apparent without ambiguity to all who interact with that being.
As far as the Buddha himself is concerned, according to the canon, those hostile to him never managed to best him in debates (thus never managed to dent his teaching) and those symphathetic to him always recognized his greatness and at many occasions decided to follow him (thus affirming his extraordinary nature). The canon tells us with this that the Buddha could, so to speak, "talk the talk" but also "walk the walk", and do this in any situation he could encounter. It is the profundity of his teachings that leads Buddhists to place their trust in the Buddha's Enlightened nature.
From this perspective making the claim was like a random person making a banal statement such as "I am a human being" for him.

If there are disciples that attain Enlightenment during their current lives, why shouldn't they also make the same declaration? There might a very simple and pragmatic answer to this. The Buddha was in the process of diffusing the Dhamma to a world that had lost those teachings since a long, long time ago. Talking about his own Enlightenment, he was giving the explanation of why these teachings were worthwhile and why he should be the one listened to among the multitudes of other teachers of the era.
What necessity has there been for practitioners to talk about their own achievements since the Buddha's death? The Dhamma has been thaught to many and codified, and the sangha was structured in a democratic and pretty much egalitarian way. People claiming Enlightenment within reason or not could/can form magnetic poles and attract disciples to themselves and away from each other, effectively sowing completely unnecessary discord within the community. It's completely unnecessary and is something that only a non-Enlightened person seeking self-interest would do.
As for the matter of the Buddha's disciples talking about their own attainments, we must not forget that
a) the Buddha was still alive, which nullified chances of successful false claims and
b) the people they were talking with were familiar with the concepts talked about and the terms used. It was a time where spiritual pursuit was seen as something perfectly real and valid in the eyes of society, and they were familiar with what that pursuit entailed. It isn't so in our modern era and Western/Westernized societies, where spiritual pursuit is not seen as something "normal" and the elements of Buddhism are foreign to the people.

This is merely a matter of the delusory power of words. When the Buddha was teaching, he used the word 'enlightenment' to express an inner state — an attitude towards the world — that he was trying to convey to others. The word had no cachet: no prestige or significance beyond the effort to describe an ineffable experience in language. He and his followers merely used it to discuss progress on the path.

As the philosophy extended and developed, however, the concept of 'enlightenment' shifted away from being a mere description of an attitude, and became more and more a label for a 'thing' that ought to be 'achieved.' Things and achievements have social value in the world. They are commodities to be bought, sold, or exchanged for other things of value; they are firm anchor points for identity and attachments. Attaching oneself to such a concept runs against the intentions of the practice.

The word 'enlightened' changed over the centuries from being used as an adjective to being used as a noun. To the extent that we can still use it in the adjectival form — as a description of a way to be — it's still a fine word. But using it as label for an attainment, a matter of pride and value, will lead both the teacher and the students astray.

The taboo of talking about or claiming ones own enlightenment is something which has been established by the monastic order. Talking about ones own attainments to lay-people or even non-buddhist is considered not done.

The only monastic rules I could find that resembles such a rule is these two:

A bhikkhu who pretends to have experienced jhāna attainments or the ariyā state without having experienced these, commits the pārājika 4. However, if the person whom the bhikkhu is addressing does not know the meaning of these words, he commits a thullaccaya.

source: http://en.dhammadana.org/sangha/vinaya/other_faults.htm

and:

Not to announce to a layman a realisation that has been achieved. If a bhikkhu announces to a layman or to a sāmaṇera, a realisation partaking with a jhāna nature or with a stage of ariyā, and this realisation has genuinely been achieved, he commits a pācittiya.

On the other hand, a bhikkhu who makes such a declaration, while knowing it to be false, commits the pārājika 4. A bhikkhu must avoid making his attainments known, even to other bhikkhus. Apart from four exceptions when they can do so, ariyās never unveil their realisations:

Under a violent threat.
Undergoing an oppressive and virulent lack of respect.
A t the time of passing away.
To reveal it to his preceptor or to a fellow bhikkhu who does a similar practice.

source: http://en.dhammadana.org/sangha/vinaya/227/92pa.htm#ch-----8

The rule only applies if one lies about attaining such, or isn't aware about what he is in fact claiming.

The second rule only applies when spoken to lay people.

Also, in the Buddhist text and stories the disciples of the Buddha also speak freely about their attainments, powers and even if they have achieved enlightenment. The other monks only seemed to protest if they didn't believe the claimant and asked the Buddha to verify, which in most cases he did.

So the rule that one must never speak of ones own attainments or claim enlightenment seems to be later added and unwritten.

I said "bragging of one's achievements is bad tone" and you turned it into "talking about one's enlightenment is taboo" :) It is very easy to fall into extremes, isn't it?

"If somebody claims to be enlightened, he probably is not" -- here is a simple explanation. Among other things Enlightenment involves shedding the mistaken identification with the illusory "I". So once Enlightenment has been revealed, it seems rather silly to say "I am Enlightened" -- because this "I" is a fiction.

Plus, Enlightenment is self-existing, it does not depend on being attained or not. The metaphor for this is sun above the clouds. Once you remove the clouds, you can see the sun, but the sun is always there, before and after. This is another reason why saying "I have attained Enlightenment" is silly.

As Chogyam Trungpa Rinpoche said:

Buddhism says that confusion has to be dispelled and that there should be realization beyond confusion. It is based on the idea of transcending the highest ideals of the human mind completely, which is called enlightenment. Once you dissolve, once your particular expectations dissolve, that is liberation or enlightenment. Since enlightenment is based on dissolving the ego and its expectations, it has been said that you cannot watch your own burial, and you cannot congratulate yourself on becoming the first buddha of the age or the first buddha of New York.

Buddha claimed Enlightenment solely in order to help his students gain confidence in his teaching. He knew very well the extent to which the statement "I am Enlightened Buddha" can be true.

Then why don't modern teachers claim Enlightenment to help their students gain confidence? As I explained in Is Satori the same as Enlightenment? and Enlightenment vs nibbana/nirvana, most people achieve "sudden insight / gradual cultivation". With gradual cultivation, you can never claim 100% Enlightenment, because in this case Enlightenment is asymptotic. All we can definitely claim is the insight.