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When you film it in UV, sunscreen appears completely black from the first drop out of the bottle, meaning it absorbed all UV. The 15 minutes might be for applying or getting absorbed before there's a chance to get wiped off.

This is a good video to highlight this [0], the sunscreen part is in the second half. Probably many more like it.

[0] https://www.youtube.com/watch?v=o9BqrSAHbTc

I always (mis?)interpreted that as a way to get you to finish putting it on before you went into the sun, so you wouldn't get burned while possibly putting it on (a group of kids, for ex) too slowly while standing in the mid-day sun.

I think the 15 mins is to allow sufficient time to dry so that you don't immediately sweat it off and/or wash if off in the pool/lake/ocean. It's got to stay on your skin to be effective.