Why did Mary need to be immaculately conceived, and why must a Catholic believe it?

We must believe it to be Catholic because the Church claims to give us the teaching of the faith definitively, as it did in defining the Immaculate Conception, so it is a requirement just as much as confessing that Jesus is the God-man. When the pope gives an infallible statement de fide, you are required to believe it. As for why she needed it: Mary needed to be free from sin so that the hereditary reception of original sin would stop at her, and Jesus would not be born with original sin. The only alternative would have been for the Holy Spirit to make Jesus immune some other way, but that is not how God chose to do it. Now I qualify this: original sin here is meant in the spiritual sense, because we are body and soul. When Adam sinned it caused a great stain on the soul, marring it, though not, as the Calvinists say, making it totally dead, as if he had no soul; the Church says it was marred to a significant degree. So the Immaculate Conception applies spiritually, to the soul, so that Jesus would not inherit original sin in his soul, but his body was still under death, which is why Jesus died on the cross and why Mary died as well, since she too is a child of Adam. We have to be careful how we slice this doctrine: if we give too much, Jesus could not have died and so could not be our sacrifice.