If you are playing a rated game, their *level* (and yours) will be displayed at the top of the game log. Their rank (that means if they are no. 1 on the leaderboard or no. 1234) can be found out another way:
Add that person to your friend list (the heart icon after the game, or by entering their name under "Friends" in the "Friend List" tab). If you now go the the "Leaderboard" tab, the tab will show the top 20 players, yourself and all of your friends. So you can find their level and their rank that way. Unfriend them afterwards if you don't want to clutter you friend list.