Firstly, you take a very big file -- say a disk image of a CD, about 700MB. Then you compress it, using a type of compression called RAR. This is similar to Zip, but can create smaller files.
When you've done this, you might have got it down to 400MB or so. The trouble is, that's still far too big for lots of people to download. Fortunately, the RAR format has a rather clever method of splitting one blig compressed file into lots of little ones. In this way, the data files are labled file.r01, file.r02, etc. There is also a "control" file called file.rar, which knows how many subfiles there are, and how to put them back together. When you open this control file in WinRAR, it automatically loads all the r01, r02 etc files (or gives you an error if they aren't all there).
As to how they work internally, well that starts simple but quickly gets very complicated and mathematical. What happens is that the compressor looks for patterns in the file. So, for example, if the sequence 1234567890 occurs several times in a file, then the compressor will say, let a=1234567890, and replace that list of numbers with the reference a. In this way, the new file is smaller than the previous one. The compressor also puts an instruction at the beginning of the file for the decompressor to do the opposite, so that you get the original file back just as it was.
Okay, so it's about a billion times more complicated than that -- maths PhDs have been done on file compression -- but that's how it starts....
EDIT: Read all about it here:
http://computer.howstuffworks.com/file-compression.htm