An indie game isn't "supposed" to be anything. Some people think of indie games as more artsy simply because indie games can be artsy. Big studios don't have the luxury of appealing to only a smaller niche audience like art games will do. They have to make back the absurd amount of money they spent making the game.
When you make an "art game," you're making a game that will only appeal to a small niche audience, which means most people will not like it. Indie games that are produced cheaply can get away with that. For example, I think "Gone Home" is a terribly overrated game, boring and reliant on cliches instead of real story telling, and lots of people agree with me. But that's ok because Gone Home didn't spend a huge amount of money on production, so it doesn't have to appeal to everyone.
I don't think any reasonable person thinks that indie games have to be art games, but they can be. They can also be point and click adventures and lots of other types of games that big studios won't touch because they won't make millions of dollars a year on them. Indies can make them because they don't have to make millions of dollars a year.