How do quantum computers find the global minimum?
Nope, that's not how quantum computing works.  That's actually a common misconception.  Sometimes people think that quantum computers work by trying all solutions in parallel and then selecting the best one, but that's not right -- that's not really how quantum computing works.

To learn more about how quantum computing works, I recommend you study Scott Aaronson's book Quantum Computing since Democritus, or his lecture notes on quantum computing.

Don't expect to find a one-sentence summary that is simultaneously enlightening and accurate.  Quantum physics is highly counter-intuitive, so you shouldn't expect your existing intuition to necessarily help you understand how quantum computing works.  Keep in mind that it's a bit unreasonable to expect an explanation in layman terms: quantum physics is tricky stuff, so your intuition from the everyday world doesn't really translate into a useful way of thinking about quantum computing.

You can also look at other questions on this site.  See, e.g., Why and how is a quantum computer faster than a regular computer? and quantum-computing.