Why are religious people against contraception?
First of all, not all religious people are against contraception. That said, many devout Catholics, Orthodox Jews and conservative Muslims are usually against contraception for two main reasons. The first is that most religions explicitly tell their faithful that they should have children, and contraception prevents impregnation. Although the Quran never explicitly exhorts Muslims to reproduce, many scholars say that this is implied.
In the book of Genesis, on the other hand, God explicitly tells his people to "be fruitful and multiply." Many other verses in the bible (both the old testament and new testament) implicitly support reproduction by equating having children with being "blessed," or with having wealth. So for many religious people, being against contraception is a simple matter of following their sacred texts, or at least following the many scholarly interpretations of those texts.
A second reason that many religious people are against, or at least discourage contraception, is that they interpret their holy books' writings on marriage and sex to mean that sex is only (or at least primarily) a means of reproducing, and should not be done simply for recreation and/or pleasure. For that reason, many devoutly religious people believe that the use of contraception is proof that the act of sex has become divorced from the act of reproduction, and is therefore sinful.
It is important to note, however, that not all religions view the act of sex for pleasure as a bad thing. For instance, Judaism as it is practiced in Reform synagogues around the world celebrates sex so long as it occurs within a marriage.