In my understanding, a claim is usually equivalent to a demand or to exercising your right to choose. In this case I think "he" was drawn to this path, not exactly knowing why. Would it be fair to say that Frost suggests "the road claimed him"? ....Now, afterwards he is reflecting on the impact of that choice. What then seemed to be an intuitive decision, turned out to have made all the difference. What I like the most about this poem is that it reminds me to let my intuition guide me and I will be all right.