The StreetPass feature in A Link Between Worlds allows you to encounter Shadow Links of other players. When you StreetPass with another 3DS owner playing the game, their Shadow Link will appear in your Hyrule. You can then challenge this Shadow Link to a battle. These battles are optional but offer unique rewards and a fun way to interact with other players.
To initiate a battle, find the Shadow Link on your map (they appear as a dark Link icon) and interact with them. The difficulty of the Shadow Link depends on the other player's progress and equipment. Defeating a Shadow Link rewards you with Rupees and sometimes a special item. There are also challenges associated with defeating a certain number of Shadow Links or defeating them under specific conditions, which can earn you additional rewards from the StreetPass Battle Guy in Kakariko Village. It's a great way to earn extra Rupees and test your combat skills against varied opponents.