These are often sampled from a clear part of the track, e.g. when the drums are on their own, or like in Fatboy Slim's 'Praise You', the vocal is taken from an old song where the intro is vocals only.
Remixes using samples are often endorsed and commissioned by the original artist so they give their multitracks - seperate tracks of drums, vox, etc, to the remix producer and they use what they need.
You can't really just seperate out the different tracks from an MP3, and if you could it wouldn't be clean sounding.
Beware that if you want to remix stuff you have to get the permission of the person who wrote the song and the people who performed it. You could end up getting sued if you release a remix without their permission.
There are many many sample CDs available with copyright-free samples - drums, vocals, riffs etc. On Reason, which is sequencing software, for instance, there are loads of drum loops from famous DJs and bands and they've given their permission for people to use them in their own mixes.
What kind of musical setup do you have at the moment?
A really good starter application is eJay which you can get for �20 and it lets you drag in blocks of samples and loops to make your own track. You can even record in your own vocals.
Good luck and watch the copyright!