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.