It means that he takes action. He doesn't think and think and think about it. If he wants something to happen, then he makes it happen.

We use "kind of" or "sort of" to "soften" the meaning of a sentence. You can think of this expression as meaning "approximately", but NOT 100% real or actual or NOT with 100% determination of focus. A similar expression might be "half-hearted" or "half-assed". He "kind of goes" for what he wants means he goes for what he wants in a loose, approximate, relaxed, indirect way. He is not 100% seriously focused on going directly for what he wants. He may go for want he wants in a roundabout way by "beating around the bush" or by going slowly or taking his times. It means he is going for it but only in a way that is described approximately accurately.