Make sure the check what level of card you are renting. most cards have abilities that only appear above a certain level. Related to this, is that the card's level is capped by the summoner you are playing. So even if you rented a higher level card, you won't get the extra abilities if your summoner level is too low. You can see if this is the case if there is an exclamation above the card when you are selecting it pre-battle. Finally, the leagues themselves have level limits, so even if you have a high enough level summoner and rented a high enough level card, you STILL may not be able to play the card at the level you rented it at.

There is lots to keep track of, but once you've been playing for a bit this will all become second nature to you.