Why are the Immaculate Conception and the Assumption of Mary such a big deal that all Catholics must believe in them? As described in this answer to a question about infallible papal statements, there are two teachings that were clearly given ex cathedra: the Immaculate Conception (that Mary, mother of Jesus, was born sinless) and the Assumption of Mary (that Mary at the end of her life went body and soul into heaven). Moreover, all Catholics are required to believe in it and not believing it means you are separated from the church.

From my Protestant perspective, I'm struggling to understand why these two dogmas are such a big deal that all Catholics must believe in them. 

I mean, I get that the Church has taken a stance on it and that they would like everyone to believe in it. But there are other statements of faith that the Church holds as true yet doubt or disbelief doesn't expel you from the Church. I'm not seeing why these matters are so important to warrant that. It seems to me that how Mary's life began and ended is between her and God and I'm not understanding how that is relevant to me and my salvation.

Why are the Immaculate Conception and the Assumption of Mary such a big deal that the popes, by speaking ex cathedra, require all Catholics to believe in them?
Catholics believe that Mary was chosen, not so that she could be the Mother of God, but so that she could choose whether or not to be the Mother of God—and that in order to make this choice in complete and perfect freedom, she was redeemed from original sin from the moment of her conception:

In order for Mary to be able to give the free assent of her faith to the announcement of her vocation, it was necessary that she be wholly borne by God's grace.
Through the centuries the Church has become ever more aware that Mary, &quot;full of grace&quot; through God, was redeemed from the moment of her conception. That is what the doctrine of the Immaculate Conception confesses ...
At the announcement that she would give birth to &quot;the Son of the Most High&quot; without knowing man, by the power of the Holy Spirit, Mary responded with the obedience of faith ..., Thus, giving her consent to God's word, Mary becomes the mother of Jesus.
Catechism of the Catholic Church, paragraphs 490–491, 494; emphases added

The Church teaches that this belief, though not always formally articulated, dates back to the beginning of the Church. As the Word of God, it has come down to us in Sacred Tradition. To deny it would be to deny that Sacred Tradition is the Word of God, or even to deny that the Word of God is Truth.
To be clear, this was part of the depositum fidei, the treasury of the faith, even before it was labeled an ex cathedra truth. But it was something that was simply assumed to be obviously true: of course Catholics would, and would have to, believe it. Pope Pius IX declared it as dogma in 1854 in the bull Ineffabilis Deus (which, mysteriously, seems to be missing from the Vatican website).
In the bull, Pope Pius stated that among his reasons for declaring this belief as dogma were to show that the Catholic Church was the one true Church, regardless of the beliefs of other churches; and to put into a formal statement something which had been believed and declared by the Church since ancient times. He asked for comment from the bishops about whether he should make such a statement of dogma, and they concurred. Therefore Pope Pius wrote:

We declare, pronounce, and define that the doctrine which holds that the most Blessed Virgin Mary, in the first instance of her conception, by a singular grace and privilege granted by Almighty God, in view of the merits of Jesus Christ, the Savior of the human race, was preserved free from all stain of original sin, is a doctrine revealed by God and therefore to be believed firmly and constantly by all the faithful.
Hence, if anyone shall dare—which God forbid—to think otherwise ... let him know and understand that he is condemned by his own judgment ...

In other words, the pope simply said that because of the antiquity and universality of the belief, it was a matter of Sacred Tradition, revealed by God. Such a belief cannot be denied without denying that God's revealed Word is Truth.
Something similar happened with the declaration of the Assumption of Mary, proclaimed nearly a century later. Again the pope (Pius XII, in his apostolic constitution Munificentissimus Deus reviewed a long history of belief that Mary was assumed into heaven, and interpretations of Scripture supporting the fact. He revealed that, just as his predecessor, he had asked the bishops of the world whether he should declare this belief a dogma, and they had responded overwhelmingly &quot;yes&quot;. He therefore stated:

We pronounce, declare, and define it to be a divinely revealed dogma: that the Immaculate Mother of God, the ever Virgin Mary, having completed the course of her earthly life, was assumed body and soul into heavenly glory.
Hence if anyone, which God forbid, should dare willfully to deny or call into doubt that which we have defined, let him know that he has fallen away completely from the divine and Catholic faith.

All this is perhaps just a long way of saying the following:
Why are the Immaculate Conception and the Assumption of Mary such a big deal that the popes, by speaking ex cathedra, require all Catholics to believe in them? Because the Church believes, and has believed from as far back as we can find, that these things have been revealed to us by the Word of God; and we cannot deny the Word without falling away from God.

Addendum: Based on a comment, it appears that the question is attempting to understand why Mary's assumption, as opposed to (for example) Elijah's1, is picked out as being so important for the Church to emphasize. The reason is simply that Mary is so much more important to the Church than Elijah is. Elijah was, certainly, a holy man, among the closest to God of all major characters in the Old Testament. But he was not so pure and holy as Mary was. There was no particular reason for him to need to be sinless; there's not the compelling &quot;appropriateness&quot; for sinlessness that there is with Mary. Similarly, though he was also &quot;caught up to heaven&quot;, it's not for the extreme intimacy of his relationship with God—a relationship that couldn't get much more intimate than that of Mother with Son. So it's much more important for the Church to emphasize the holiness of Mary than that of Elijah.
1The original poster chose Moses as an illustration; but Moses died and was buried, whereas Elijah was taken up to heaven.