The trick is to realize that "same birthday" isn't a property of people, it's a question about pairs of people.

A group of 23 people has 253 unique pairs. So if I ask, "In a room with 253 pairs of people, what are the odds that in one of those pairs, both have the same birthday?" Now around 50% seems a lot more intuitive.

It's unintuitive because the number of pairs increases quadratically with the number of people.