As with any religion, different people are attracted to Asatr for different reasons, but I'll try to touch briefly on a few of the reasons that I've heard said more than a few times. Asatr values ideals of individual honour and virtue. It is the natural religion of our ancestors. It is a very simple, straightforward, natural set of beliefs, that nevertheless contains complex mysteries and hidden knowledge that would take many lifetimes to completely understand. Personal experiences with the gods.
Many people tend to have visions, dreams, or other esoteric type experiences with the gods.
Experiences that years later, still seem powerfully real and actual. I like to say that Asatr is the only religion which requires NO faith. Asatr encourages us to be who we are, and to grow toward being the best that we can be. The gods expect our civility and respect. They desire their people to be free and mature, not dependent and helpless.