To grasp it intuitively, I think of it like this.

With the first person, you have 1/365 chance the birthday will be on any given day.

Each person you add to that adds not just another person but also another day that can be a match.

After two people, you still don’t have a match but now you have two days. The third person can match either of those. That’s a lower bar than person #2 had to meet.

By the time the 15th person walks in, the question is: “what are the odds that you share any of these 15 days as your birthday.” And remember, it’s not that that person’s odds are 50%. It’s everything from the original 1/365 chance on up to that fifteenth person, cumulatively, that has a 50% change of a hit.

See how this already sounds a little more likely than just narrowing in on the final final result of two people having the same birthday? The way the problem is phrased makes it sounds like more of a bullseye than it truly is.

So I think part of it is just difficult to grasp intuitively, but it’s also phrased deliberately to throw off your intuition.