They provide me with a (sometimes very sad) story. When I started reading VNs, I was at the brink of loosing interest in the story of games in general, due to the fact that I believed that the story of "todays" games had nowhere near the strength and power of the stories of older games, such as FF VI or Lufia II. Whenever some gaming magazine or person says that games such as "the last of us" are the first games which really pay attention to story, I mentally facepalm.
VNs provide (as Rusanon pointed out) a story that usually is very "unrestraind" and showing whatever the writer wanted to convey. In the west, whenever you make a game which has even a very little thing that alludes to, say, sexual things (Mass effect, anyone?) media will instantly make it into a huge deal. So it's the freedom that VNs have as a medium that is much of it's appeal to me.