This is all about sex, of course -- in particular, the question of purpose: what it's for.
I will preface my answer by saying that contraception is not directly addressed in the Bible, so my answer is going to be primarily theological rather than biblical.
Catholicism says that sexual acts, to be moral, must have two purposes: that of expressing and confirming the love between the couple and that of procreation -- conceiving children. Contraception prevents the procreative purpose, so the sexual act exists solely for the former purpose. According to the Catholic understanding of the purpose of sex, therefore, it is immoral. Similar arguments can be made about homosexuality, masturbation, oral sex, etc etc. Note that before the publication of Humanae Vitae (the encyclical that effectively bans artificial contraception) the Pontifical Commission on Birth Control actually had a majority report in favour of revoking the ban.
As for Protestants, the case is (inevitably) rather complicated. Some Protestants do indeed see contraception as inherently sinful. They say that planning families is for God to do, so contraception is obstructing God's purpose, and often also echo the Catholic position about the purpose of sex.
Many Protestants, however, believe it possible to separate the two purposes of sex (love and reproduction) -- so it is possible to have a non-sinful sexual relationship without the prospect of children. For instance, the 1662 Anglican Book of Common Prayer has specific provision for couples who are beyond child-bearing age. If a sexual relationship can exist without the possibility of reproduction, and if you believe that God's will can be exercised through the working of human free will, it's not hard to believe that contraception can be non-sinful.