You have to understand that in the US, the right is religious and the left isn't. This gives them different worldviews.

The religious perspective is generally that the fetus already has a soul and thus is a person - and killing a dependent is murder no matter how taxing their life is to your body. So while the right likes rights, they can't abide what they see as child murder.

The left doesn't believe in souls, generally. They take the more biological view that you aren't really a person until fairly late in development. The fetus is just a fetus, not a person until very late - and having a heartbeat or neurological activity makes the fetus a living animal, but still not developed enough to be a person. So it's acceptable to remove it if it's not very developed.

Personally I take the latter view, but if I was raised to believe that fetuses had souls I can imagine being pro-life instead.