It's a very simple process, but - of course - you do have to have the right software. Two popular programs for this are Sound Forge and Syntrillium's CoolEdit. I'm sure there are others, but I use both of these quite a bit. Basically all you do is load the MP3 into the software, where you can view it, play it, highlight the parts you want, or the parts you don't want, and then cut, paste and save the new part.
Both pieces of software have similar functions, but CoolEdit is by far the more professional of the two, and was at one - possibly still is - the chosen tool of many radio stations in the UK.
You might be able to find Sound Forge 4 as a download and it will do exactly what you want.