It allows you to visually flag the pets that you want/need to level. Also, if you have carry teams saved in rematch you can set that team up to automatically include pets from your level queue. Once they are level 25 or you manually remove them from the queue - they are replaced by the next pet in the leveling queue.
You can add a leveling pet to a carry team by right clicking in the box for the pet on the Rematch gui and selecting "Put Leveling Pet Here".
If you save the team w/ that option selected for a slot, it will always include pets from your leveling queue automatically for you.
Hope that helps