The Rarity Exchange will exchange your maxed cards for one of a higher rarity once you fill-up the meter. The meter increases automatically each time you receive a copy of a maxed-player card.
For example, filling the Common card meter by obtaining the indicated number of Common cards will award a Rare card.
At this time, it doesn't show you which card you earn - a random card from the specified rarity will be added to your player totals. This can be a card you already own, and isn't necessarily always a new card.