Primarily the reason math teachers ask you to show your work in high school is to prepare you for
advanced algebra and
calculus. Perhaps it does slow you down, but at the same time, if you can't
prove your work, it will be thrown out. For all anyone else knows, you got the answer by
magic. This is a basic tenet of science and mathematics:
reproducability. If someone else can't do the same thing you did and get the same result, then you (or anyone, in the case of new research) don't understand what you're talking about, or you can't pass on that understanding to the next person, thus making your work pointless.
Also, it's worth noting that it's much easier to understand where you went wrong if your entire process is written down. This means that not only can you go back and fix your mistakes, other people can too if they find them. Think of it as mathematical open source.