Writing a mathematical proof is like explaining math to an alien. Imagine: you’re sitting at the table doing your math homework when the family alien (for this is the 80s and every family had one) comes to sit beside you. He asks you what you’re doing and you show him your work. Now, since he comes from a distant planet, he is unfamiliar with Earth methods and begins to ask lots of questions. You find yourself stopping after each new mathematical maneuver to explain the reasons why it is possible. The alien seems to be following along, so you continue in this manner. Eventually, you’ve reached the endpoint of the problem, and the alien can see and understand exactly how you got there.
It’s also like playing an adventure game. You’re plopped down in a specific place and given nothing but a few select pieces of information about your current surroundings (“This is a pirate ship. It’s captained by Blackbeard Joe.”) and an objective (“Rescue the princess.”). From there, you wind your way around the ship, looking at everything, uncovering more of the map, pocketing the tools which seem useful, and gradually working your way towards figuring out how to achieve your objective. You backtrack a lot, and you also look forward. You find out where the clues you have fit into the ones you’re just now unearthing, and soon everything falls into place. You find Blackbeard Joe, rescue the princess, and even manage to escape with his treasure.
Direct mathematical proofs are a form of deductive reasoning. You start with what you know, and figure out systematically what you can deduce from it. Then you take that new piece of information and add it to your list, and repeat the process again. You move one step at a time from the known to the unknown, until finally the whole thing clicks into place and you see where you were meant to arrive.
Completing an indirect proof, though, is like playing Devil’s Advocate; you prove your point by disproving the opposing one. Imagine: you’re having a discussion with a friend about a hotly debated topic. To prove your point, you begin “Let’s suppose that what you’re saying is true. If that were the case…” You go on to expand upon their starting point until logic brings you to a statement that is blatantly absurd or incorrect. You use this absurdity to prove to your friend that their opinion, the starting supposition for your argument, is false, thereby proving that your opinion (the opposite one) is correct. If you are asked to prove indirectly that two lines are not parallel, you begin by supposing they are. You work logically through the proof from there, and when you reach an inconsistency (such as 3 = 4) or a statement that contradicts one of your givens, you know that your original supposition (that the lines were parallel) is itself false. The lines are not parallel. Congratulations, you’ve proved your point.