What is the Exchange Booth?

You need the new games, as those are the only way to get Plaza Tickets (you get them for accomplishing various goals in each game - they are like rewards for gaining achievements).

Once you have at least one of the new games, you can start earning tickets, and then can exchange them there for various outfits or hats for your Mii (they change apparently daily, but I don't know what clock they use.)

Yes, it does require you to buy a new game. In the "Intro to Game Purchasing" it says:

When you play them, you can find new Mii accessories and earn plaza tickets by achieving certain goals.
_{Pleas not that plaza tickets aren't available in Find Mii, Find Mii II, or Puzzle Swap.}