In many other games, Streetpass can get you special items or other abilities. Does Pokemon X/Y have anything special with Streetpass?
StreetPass provides you with an amount of PokéMiles (PM) for each person you've met who is also playing a StreetPass enabled X or Y. PokéMiles can be traded in for items at the Lumiose City Pokémon Center. Alternatively, you can redeem them at the Pokémon Global Link website via the Game Sync feature of the PSS; the prices there are lower than in Lumiose, and the items you select will be delivered to the Pokémon Center of your choice. Eventually, you will be able to use your PokéMiles to play mini-games called "Attractions" on the Global Link website and win items, though at the time of this writing, no Attractions have been released yet.
To get rid of that green StreetPass light on your 3DS' homescreen, you have to go to the PSS (on the touch screen) and open up the Holo Caster (not to be confused with the Holo Caster item in your Bag). Once you're there, hit "check announcements." Then hit "Streetpass News" and you'll be notified of how many people you've StreetPassed, how many PokéMiles (PM) you got for that, and your total PokéMiles.
You can also earn PokéMiles by trading or battling with other players online (this includes Wonder Trade and GTS trades). The number of PokéMiles earned (either via StreetPass or online interactions) is based on the distance between the locations set on both players' systems; international interactions are worth more than local ones.