Arenas are stages in which players will battle their opponents. As a player wins and collects enough Trophies, they will advance to the next Arena. Doing so will change the environment, as well as allow players to collect new Cards only available for that stage.
But beware! Losing enough Trophies can put you back into the previous Arena! When this happens, you will not be able to earn Cards from the Arena you were bumped out of!