The solution: an old shaving razor and some rubbing alcohol.
Read on for more hand-holding.
Step 1: Marvel at How Dirty Your Mouse Is and Gather Some Sharps
Then check your inventory of sharp objects: an old multi-blade disposable razor is perfect for the job, but any relatively sharp knife works too (e.g. X-acto knife, Swiss army blade).
And just in case you were wondering, the mouse works through electromagnetic induction.
Step 2: A Shave...
Preferably one with more than one blade (this leaves a nice, clean surface on the felt). Shave the bottom like you would normally shave, well, yourself. Don't push down hard and don't angle the razor too low or it will take off chunks of the felt and cut into the plastic along the edges of the mouse. Go slower along the edges to avoid nicking the plastic. Also go a little slower over the sticker in the center to avoid cutting the edges of the felt.
Any other type of blade:
Hold it perpendicular to the surface of the felt and scrape LIGHTLY. If the blunt side of the blade is about the same thickness as the cutting edge and you can use that to scrape without cutting yourself, do so (the non-cutting edge of an X-acto knife works very well). You don't want to scrape too hard or at an angle because it'll take off bits of the felt.
Step 3: ...and a Haircut.
Then, depending on how obsessive-compulsive you are, you should be done with the bottom of your mouse.
Step 4: Finally: Clean Off Your Hand-filth.
Your mouse might not look that dirty, but trust me, it is (see picture). Another way to disgust yourself is to try this with your game controllers.
Well, this is as clean as I care to get my mouse... if you're extraordinarily obsessive, you can clean the cracks with a toothbrush or something.
Have fun re-dirtying your now-clean mouse!