In a perfect world, every file on your hard drive would be stored in one clump of data. In tech-talk, we say that each file would be stored on contiguous sectors on your drive. It's quick and easy for Windows to open a file that has been stored on contiguous sectors because the read-write heads have to move a tiny distance from sector to sector.
Over time, files become fragmented. As you install and uninstall software, and as you update and save files, they tend to be stored in chunks all over your hard drive. This causes two problems. First, it slows down your computer because the read-write heads are moving wildly each time you read or write a file. Second, it wears out your hard drive. Disk Speedup defragments the hard drives and arranges fragmented files sequentially. This solves both problems.