Some games store all your game data on their servers. These are generally the horrible click once a hour to get some resource and then they charge you if you want to speed things up.

Or it can be as simple as they require internet access to show you ads.

My rule of thumb is to pay for apps and not download ones with in app purchases.

Some also have it as a form of copy protection. That is why many gamers are really against DRM (Digital Rights Management); it barely slows pirates down, but things like mandatory internet access do inconvenience legitimate end-users.

@jerv Is spot on. As an aside: best anti-piracy technique ever.