If both the Orthodox and Catholic Church affirm salvation by grace through faith, why did the Protestant Reformation happen?

CuriousDannii's answer is quite good, and well researched. I'm adding an answer that is perhaps more conceptual and experiential.

Firstly, there's a number of reasons for the Reformation on grounds other than salvation by grace through faith: e.g. originally the abuse of indulgences (not strictly doctrinal), then Papal supremacy, the role of tradition in informing doctrine, primus inter pares of the Pope, the role of the saints in heaven.

Beyond those are a host of reasons that all Protestants don't entirely agree on either (but have the freedom to do so).

In terms of the doctrine of salvation by grace through faith, it comes down to the matter of only. (sola fide) Orthodox and Catholics believe that you are saved by grace (God's redeeming action), but this does not mean that you are excluded from participating in this. The doctrine of total depravity is the key point of difference.

1. Protestants, generally, would believe in total depravity, and in fact that it is not even possible for anyone to do good works, not just with the help of God, but in fact it is the exclusive work of God.
2. The Catholic side would say that this is removing God from His role in nature, and underpinning what is a natural good. Protestants would say that God is not as involved in nature (being the natural world), that making material things holy is idolatry, and argue that Catholics tend to just tick boxes and have works without it being rooted in faith (which whilst not uncommon, often tries to be corrected by priests in the Catholic Church).

Full disclosure: I'm Catholic, previously Calvinist (5-pointer even), and completely believe that there are many things in this life that can be good.
Please comment if you think I've been biased or unfairly painted either view. I acknowledge that Protestantism in some instances has become quite Catholic such as the Oxford movement in the Anglican Church or the Arminian rejection of predestination.

Hope this helps, wishing you grace for caring about this, and my best wishes resolving this question.

While it is true that one can say the Eastern Orthodox proclaims "salvation by grace through faith", the eastern understanding of what exactly salvation is departs radically from that of western Christianity. This was actually made clear in the local Orthodox Church tract another answer referenced, but seems to have been overlooked (p.78-79):

The problem is that Orthodox and Evangelicals do not use the word saved in the same sense. This means we are talking about different things. In the evangelical understanding the satisfaction theory of atonement is assumed. It presupposes that the difference between the saved and the damned is the attitude of God toward them, not any inherent quality of their own. It also presupposes that our state of being guilty can be changed in an instant. For an evangelical, to be saved means to be declared “not guilty” by God.

From the eastern perspective, western Christian confessions inherited this largely juridical interpretation of salvation, wherein Christ's death on the Cross was a key event in human history that finally accomplished retribution for man's disobedience in the Garden. While the eastern Church might have agreed with some of the motives of the Reformation (e.g. rejection of the notion of Papal primacy), it also sees that the basic premises of penal substitutionary atonement - perhaps most aptly articulated in the 11th century by Anselm of Canterbury - had been carried over.

The Archangel Gabriel announced to the Virgin that Jesus would save His people from their sins (Matthew 1:21). Elsewhere we read that the Son of Man is come to seek and to save that which was lost (Luke 19:10), that He is not come to destoy men's lives, but to save them (Luke 9:56), and that He came not to judge the world, but to save the world (John 12:47). Commenting on these passages, the late Archbishop of Dallas, Dmitry Royster, wrote:

What does "being saved" mean? From what sins do men need to be saved? Since sin in the Greek original is hamartia, literally "failure" or "missing the mark", we have to conclude that man's sin consists fundamentally in his missing the very point of his existence (although for some Christians, salvation has been reduced to nothing more than escaping the punishment of hell).¹

A further contrast of Eastern Orthodox and western Christian (Roman Catholic/Protestant) fundamental beliefs concerning the nature of sin can be found in a later edition of M. Pomazansky's Orthodox Dogmatic Theology, where the editors comment:

Roman Catholic teaching on inherited guilt is based on the works of Blessed Augustine,² who wrote "Even of believing husbands and wives are born guilty persons ... on account of original sin" ("Treatise against Two Letters of the Pelagians, Ch. 11); "The fault of our nature remains in our offspring so deeply impressed as to make it [our offspring] guilty" ("On Original Sin", Ch. 44); "Inasmuch as infants are not held bound by any sins of their own actual life, it is the guilt of original sin which is healed in them by the grace of Him Who saves them by the laver of regeneration [i.e. in Baptism]" ("On the Baptism of Infants, Ch. 24); "Until then, this remission of sins takes place in the offspring, they have within them the law of sin in such manner that it is really imputed to them as sin; in other words, with that law there is attaching to them its sentence of guilt, which holds them debtors to eternal condemnation" ("On Marriage and Concupiscence", Ch. 37). The concept of inherited guilt was affirmed at the fifth session of the [Roman Catholic] Council of Trent (1546), which, in defining the Roman teaching on original sin, referred to the "guilt of original sin." In the 17th and 18th centuries, some Roman Catholic theologians continued to develop Blessed Augustine's teaching on inherited guilt. However it was in Protestantism rather than Roman Catholicism that the doctrine of inherited guilt (also known as the doctrine of "imputed sin") was given its most extreme formulations.³

<sup>1. St. Paul's Epistle to the Romans: A Pastoral Commentary (St. Vladimir's Seminary Press, 2008), p. 33.
2. St. Augustine is venerated as a saint in the Eastern Orthodox Church, but he is also acknowledged to have committed some theological errors during the course of his vigorous condemnation of Pelagius. A fairly detailed explanation of this can be found in S. Rose, The Place of Blessed Augustine in the Orthodox Church (St. Herman of Alaska Brotherhood, 2007). Some of these Augustine himself recanted later in life.
3. Orthodox Dogmatic Theology (3rd ed.; St. Herman of Alaska Brotherhood, 2005), p.165n.</sup>

I'd like to focus on the following part of your question because it is not answered, yet:

If ..., why did the Protestant Reformation happen?

This question implies that the main reasons for the separation of the Catholic and the Protestant Church were of religious nature.

When learning about Luther's time in history lessons in school here in Germany, we learn about the reasons that played an important role: Many (or even most) of the reasons for the separation were not of religious nature.

The historic background in Germany is little known outside Germany and it is rather difficult to explain.

One reason for the success of the Reformation (not for the Reformation itself) in Germany was that most monarchs in Germany were against the pope for economic but not for religious reasons.

You could also look at the Anglican Church:

The reason for the separation was that King Henry VIII wanted to get divorced and marry another wife.

According to the German Wikipedia, the pope first wanted to allow the divorce but the reason why he did not allow it was fear of the king of France.

Fear of the king of France is obviously not a religious reason.