I don't know anything about any incentives to download the app or if Google pays them if they have a certain threshold of downloads.
But it will be a big plus for Splinterlands if they have a good app, knowing that it's a game easily playable on mobile during a couple of minutes of free time.
Well, I consider the availability of the app as a way that should generate income for the game. I mostly play on web and unless the site is down, I still prefer to play on web then on app, but yes, this is good to have multiple access to play the game.