Why does restarting a computer fix things?

Basically because anything that's got in a mess gets the chance to start over. Imagine you're making toast and you burn it. Throwing it away and starting again is one way to fix that problem and will always work out better than scraping the burnt bits of toast off.

One of the major reasons your computer slows down is that its Random Access Memory (RAM) is being used. The operating system, as well as the programs you're running, all use RAM. However, there's only so much of it, and it can only be accessed so fast. If your computer is trying to use a lot of RAM (often more than is availablee), it slows down. It needs to create files extra swap files on the hard drive to act as extra, but less efficient, "RAM". This, among other things, makes your computer slow down.

Closing some programs should free up RAM space, but memory leaks may have occurred. That means that the program may have accidentally taken up RAM that it didn't/couldn't free up when it closed. "Ahhh" you say, "it's going to eat up all my RAM!" Nope. If you restart the computer, all the RAM is cleared out. You've got more avaliable RAM, so your computer can run faster.

There are other problems that could be fixed by a restart, too. For example, if a program somehow begins to use a huge amount of processor cycles (each cycle consists of a calculation, and all of these calculations make your computer "compute", aka work). When the computer is restarted, the control of the processor is unconditionally given to the bootloader, and then it's handed off to the OS, which can start from scratch. It's no longer being dominated by the greedy program.

Yet another possibility is that the computer was overheating. Overheating, simply put, isn't good for the computer. Turning a the machine off and leaving it to cool for a few minutes couldn't hurt. In fact, some (if not all) computers are set to shut down if they reach a certain internal temperature.

In summary, a restart puts the computer into a state where the right software is controlling the right (possibly cooler) hardware, in a state this is already known to work right.