Why the commonly accepted "immutability of Jesus teaching" is so important? How to understand its goodness, meaning, rational content?

The unchangeability of Jesus' teachings is due to who he is. What he taught was the word of God, indeed, he stated that he did not say anything of his own but only what he heard the Father saying.

"...he that sent me is true; and I speak to the world those things which I have heard of him. They understood not that he spake to them of the Father. Then said Jesus unto them, When ye have lifted up the Son of man, then shall ye know that I am he, and that I do nothing of myself; but as my Father has taught me, I speak these things." John 8:26

After Christ was lifted up by resurrection from the dead, his promise to those who believed him proved true: the Holy Spirit came to remind them of what he had said, and to guide them into all truth (John 14:25 & 15:26). Truth, that is, as to what Christ taught - not every fact that can be known in or about the cosmos! Now, that is the way Christ's followers look at it, because they look to him. Needless to say, the world does not see it that way at all. They will not look to Christ for they are looking to themselves, their own understanding, and the way the world is, as they see it. Therefore, as physical things in this material world are seen to develop, or to be changed by human interventions, and human teachings keep changing, the human mind expects Christ's teachings to change in harmony with what is viewed as improved human intellect and understanding. Christ's teachings can be adapted to our 'modern' way of thinking, is the idea. And when changes, or 'developments', in Catholic teaching are noted over the centuries, this is viewed as a theologically acceptable process.

Well, this is where the starting point for viewing the immutability of Jesus' teachings has to be rediscovered. As one theologian put it:

"God is holy in character and is righteous in all His ways. An absolute standard is affirmed. There is no 'situation ethic' with God: no bending back of integrity to fit what we may deem to be convenient: no compromise. In fact, the entire thesis of the Bible stands or falls at this very point. If the moral confusions of a pluralistic, multi-faith, consumerist society may be validated by adapting principles, willy-nilly, to every wandering star of fashion, then surely the Bible has nothing but its own pennyworth of suggestion to throw into the kitty of daft ideas!" Portsmouth Baptist Church mag. P2, January 1999

God never makes any changes to either his perfect will or his moral law, which stands for all time, whether we like that or not. Those who wish to be a law unto themselves always have, and always will, seek to tinker with the perfect law of God, to adapt it to their way of seeing things, to their liking, to their view of the world. Because they have started with their own thinking and desires, and not with God's immutable statements, the situation (to lob a bit of Latin at it) becomes one of ignotum per ignotius - The unknown by the still less known.

Human thinking is 'the less known'; God is unknown to those who impudently believe their knowledge can even equal God's, and if they can improve on their knowledge, then so ought God - to keep up with them? Is that what is really in their minds? That they are so brilliant in thinking and understanding, that God's teachings should change, to adapt to modern understanding of humanity? Are they just looking for an excuse to twist what Jesus actually taught to accommodate their vastly different moral and legal world? Is a divine 'rubber stamp' sought, to justify their explanations that are more obscure than the things they try to explain away?

The reason why the immutability of Jesus' teaching is so important is that it is the very teaching of God. That is because he is the Son of God, sent from God into our sinful world, to save that world of humanity, lost in their vain thinking. Nothing has changed. Except the wriggling of the worm caught on the hook of God's perfect law and perfect will. Catholicism will not concede that Christ's teaching can be changed, nor will Protestantism (as official religious groups). Individuals within do challenge that and offer different interpretations, or even their own teachings, but to go so far as to say Christ's teaching can be changed is to step outside the bounds of Christianity.

If God's immutability has no value, as you ask, then we are in a godless system of capricious, random circumstances, where no purpose can be traced in the random flow of atoms.

How to understand it and its transcendental significance? Because of the immutability of God, Christians can trace his constant purpose and unchanging will, to rise above the science of existence to the spirituality of it.