I like this analogy:

Imagine a massive wall of lockers. Each locker is 1mm by 1mm, and the entire wall of lockers is a square 2 light years on each side. When you choose a private key, you pick one of these lockers at random. When someone sends you bitcoins, there's some magical inbox which puts the bitcoins into your locker without telling the sender anything about the location of your locker.

The lockers don't have locks. If someone knew the location of your locker (ie. your private key), then they could just go take what's in it. Similarly, it's possible to choose a locker at random and find that someone has used it already at some point in the past. But there are just so many lockers that in reality it's never going to happen, even if humanity devotes all of its efforts to searching through all of the lockers.