Every 20 total points of damage dealt to and points of life lost by your opponents by effects you control, you may choose and set aside a card that you did not put into your deck—
That is, a card currently in your sideboard—then cast that card anytime you would normally be allowed to cast it without having to pay its mana cost.
For spells that have additional costs to cast, like sacrificing a permanent or kicker, or spells that have an X in their cost, you pay those additional costs or X values as normal. Whew! That's a mouthful!
What this rule really gets at is that you want to be attacking to deal damage, dealing damage with burn spells and life-draining effects, or otherwise actively damaging your opponents as often as possible. It's something you probably want to do anyway, but you get a reward for it too. Neat, huh?
While the cost of attacking into an opponent is often a risky situation, the Rumble Rule changes this to an advantage—if you hit enough times you can bring out something absurdly good or difficult to cast, or both, if you happen to have something like Progenitus in your sideboard. Instants and sorceries that are off the colors of your deck but would otherwise be great to play, like Sleep, can be used to surprise or switch up a situation. You can even grab a land or other mana fixer you need to cast the cards in your hand; fixing some mana troubles can mean the difference between sitting duck and sitting pretty!
And, perhaps best of all, that awesome card you picked up for one of your decks (like that Serra Angel from before, remember?) can even come out to play!
Sounds like an interesting variant to try at the shop some time