One side starts with the belief that sex is something you can totally choose to do with a consenting partner. If this is true, then it's normal for people to have sex when they're married, dating, hooking up, whatever- just decide what your own boundaries are, and that's great.
If this is true, then women HAVE to have access to abortion. Because who knows whether you'll be in a position where you can deal with being pregnant and having a kid?
And it doesn't matter whether the fetus is a person or not. We don't force people to donate their kidneys to save someone else's life- why would we force a women to allow another person to live inside her body like that? It's not her fault she's pregnant- she's just living life like a completely normal person, ie having consensual sex when she decides she wants to.
The other side starts with the belief that sex is only for marriage. If this is true, then when a woman gets pregnant, she'll have her husband there to support her, and they'll be able to deal with the pregnancy and raising a kid.
And if other people want to have sex outside of marriage, okay, they have the freedom to do that if they want, as long as they're not hurting anyone else. And abortion IS hurting the unborn child.
This position often gets criticized as being all about controlling women or judging them for having sex. No, that's not what it's about. It just says if you get pregnant, you have a responsibility toward that unborn child.
And the two sides argue past each other- emphasizing that "life begins at conception" when the pro-choice arguments don't hinge on when life begins, or that "a woman should have control over her own body" when the pro-life side is actually concerned with control over the fetus's body.
People are never going to see eye-to-eye on this because they have fundamentally different understandings of the role of sex, and that's a religious belief. Anytime you say "sex SHOULD be [fill in the blank]", that's a statement that cannot be proven or disproven. It IS a religious belief, because it deals with how you see the world and how you think the world should be. Even if you don't follow any specific religion, you still have beliefs that can't be proven one way or the other.
And here's the problem. Everyone should have freedom of religion- but that freedom ends when it hurts someone else. One side argues that pro-choice religious beliefs about sex hurt unborn babies. The other side argues that pro-life religious beliefs about sex hurt women.
Any policy we make about abortion is going to be "imposing religious beliefs" on someone. There's no way to be neutral.
(And I need to throw in a disclaimer about how I'm making some huge generalizations here, not everyone with an opinion on abortion is going to nicely fit into these two categories, etc etc. But I think I'm summarized the essence of each side.)
What should we do? I have no idea. But let's at least start by attempting to understand the perspective each side is coming from.