I keep hearing people say "tax the rich" but how do the rich not pay taxes? (sorry for the dumb question, I don't know much about this)

That's a question that could be answered exhaustively perhaps by a book, let alone a post. Anyway, most common routes on top of my mind:

1) hiding assets out of the govt's jurisdiction
2) moving businesses headquarters in tax havens, taking advantage of trade agreements between differently taxed jurisdictions (fiscal optimization, kind of)
3) declaring less than they actually make, or even straight out faking/omitting invoices
4) selling or transferring possessions to a trusted person, so it no longer appears as yours, while actually you still own that possession. Most trustable person would of course be a dead person