You know you have a stable home life and are ready to adopt, but they don't know that. With same-day adoptions at least some people would adopt on a lark ("hey honey, I got us a puppy!"), which usually does not go well for the pet.
Further, the shelter might be using that time to check with your landlord (if you rent) and/or your vet to make sure that you're not just confidently saying "oh sure, it'll be fine" as people are wont to do when excited by a new possibility.
Further, especially if this is your first pet, there may be some things you need to do to ready your home that only come up in discussion with the adoption counselor.
Finally, if there are family members who are not with you when you visit the shelter, they might want to hold the pet while you arrange for them to visit. My shelter required all members of the household to be present; I think this is in part to verify that everybody agrees to adopting a pet and in part to observe everybody interact with the pet in the play room.
If you and Rover are meant for each other, the shelter wants to help you have a happy life together. In the grand scheme of things, if that happy life together has to wait until tomorrow or the next day to start, is that really a problem?
By the way, some shelters allow you to fill out the application online, so they can do all these checks in advance. That's what I did with my two cats; once they reviewed the application we scheduled an appointment, my husband and I went there, and we brought them home the same day. If making two trips to the shelter is a problem, check to see if you can apply in advance.