Discounted outs are outs with those cards subtracted that would improve the hand but not improve it to the best hand, or that might improve an opposing hand more.
Example (Texas Hold'em):
The player has a draw to a straight. Any ace or six on the turn or river will complete his hand. Those are his outs and he can start out presuming to have 8 outs; four aces and four sixes.
But what if an opponent had this:
In this case, the ace or six of hearts would give his opponent a flush. These two outs are not clean, and so they must be discounted.
An opponent might also have this hand:
In this case, any six would give his opponent a better straight, meaning that only the four aces are clean and so he must discount 4 of his original 8 outs.
Whereas it is difficult to say, in game, what an opponent might have, there is always a certain probability that he has one of the better possible draws. The more players are in the game, the more likely it is that one of them has such a hand. The more playable are the cards that would complete your draw, the more likely are players to hold them. Hence, the strictness with which one must discount one's outs are contingent on these factors. The discounted outs take into account the possibility that even though the draw may complete, it may not win the hand. They keep the calculation of one's chances to win with odds and outs simple.