Imagine your problem is to write Hamlet by Shakespeare. One way to write Hamlet by Shakespeare is to enlist many monkeys who then type at random (but with the property that no two monkeys type the same thing). Each monkey also has a special "done" button that they press when they're done writing. Some monkeys never press it and write forever.

So you instruct each monkey to type one key and then you check each monkey to see if they pressed "done". If they pressed "done", you check if they wrote Hamlet. Otherwise, you continue and have the monkeys each press another key.

Since there are infinitely many different monkeys, so you can't enlist them all at once, because then checking after each key press would take you infinitely long! This is why you play that game with at the first step you enlist one monkey, then two, then three, etc; it ensures at each step there are finitely many monkeys to check.

Of all the possible monkeys, one monkey will write Hamlet exactly and then press "done". This scheme finds that monkey. Similarly, for Turing machines, there will be at least one (technically, infinitely many) that solves your problem. You just have to figure out which one it is by doing the enumeration that stephencanon detailed. If P = NP, that enumeration process can happen in P time. Keep in mind that all problems in NP have polynomial time verifiers. So you can always check a solution (i.e., checking if the monkey actually wrote Hamlet) in polynomial time.