Why should I use asserts?

I never got the idea of asserts — why should you ever use them?

I mean, let’s say I were a formula driver and all the asserts were things like security belt, helmet, etc.

The tests (in debug) were all okay, but now we want to do racing (release)!
Should we drop all security, because there were no issues while testing?

I will never ever remove them. I think most of the guys that claim that removing something comparable to asserts never profiled their code or the asserts were absolute displaced.
I’ve never seen any real performance advantage especially regarding the 80 / 20 rule.

So, am I missing the point somehow, or could anybody tell me, why I should use asserts?
By the way, I’m using unit tests.

Why should I use asserts?