How do quantum computers find the global minimum? If I understand correctly, quantum computers work by trying multiple methods of solving a problem simultaneously, using quantum superposition. However, if you try to "look" at a superposition-ed atom, you will find either one or the other-1 or 0. What I've heard is that this is random.

So, if you compute all these solutions, how do you know which is the best without "looking" at it? Wouldn't the result just be random, and you can only see the result of one answer? Now, I've heard that this can be solved by quantum annealing, but I don't really understand it. I'd like to know how this works in fairly layman-ish (It's a word) terms.
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.