Baby swap is for rides that have a height restriction. That way parents with a tiny child (like myself) can each ride while the other parent holds the baby, but they don't have to wait in line twice. For example, on RnRC, my husband can wait in line (or get a fastpass), ride RnRC, then when he gets off, I can ride it without waiting in line. It depends on the attraction how they do it, whether you go in a back entrance or through the fastpass line, but either way you save the time of waiting in the regular line again.
You could accomplish the same thing using a fastpass for each parent, but of course that won't really work if you have other kids, just for single kid families. That's the really nice thing about it though, that you can just get one fastpass per ride, instead of one per parent. (Again, only really applies to single kid families.) So you can get 1 fastpass for Thunder Mountain, 1 fastpass for Splash (both at the same time and then a babyswap pass for each. That way both can ride each ride, but you can get the fastpasses at the same time, no need to wait till the time printed on the bottom of the ticket.
I have to admit though, we've sorta done what brunoflipper mentioned. We've gotten a baby swap pass for a ride before, but after my husband rode we had a fastpass or ps or something so there wasn't time for me to ride, so we'd save the baby swap pass for later (no time on them like a fastpass). Some of those time I wouldn't feel like riding (I'm really not a thrill seeker, but sorta force myself to do it to overcome the fear) so my husband would ride again instead of me. We never did it with that intention, but sometimes it worked out that way, and it was much easier than waiting in line again or getting another fastpass.