If your combat strength totals more than the monster’s – note that monsters win ties! – you kill it and go up a level (two levels for some big monsters). You’ll also get the number of Treasures shown on its card. Discard the monster(s) and any other cards played.
But note: someone may play a hostile card on you, or use a special power, just as you think you have won.
When you kill a monster, you must wait a reasonable time, defined as about 2.6 seconds, for anyone else to speak up. After that, you have really killed the monster, and you really get the level(s) and Treasures, though they can still whine and argue.
Sometimes a card will let you get rid of the monster without killing it. This is still “winning," but you don’t get a level. Unless the ability says otherwise, you don’t get the Treasures, either.