Where does the traveller find himself? What problem does he face?

While a literal reading of the poem suggests the problem is choosing which path to take, the speaker’s actual conflict is more existential. The speaker deliberately chooses the path which he determines is “less traveled by” (line 19).
This suggests that the speaker considers less conventional, more difficult paths in life to be better than what the average person might choose. This is reflected in the last line of the poem, where the speaker says this choice has made “all the difference.”
Of course, the speaker is contemplating the future significance of his choice, since the beginning of the final stanza indicates he is merely postulating (“somewhere ages and ages hence”). This implies that the speaker wants to make sure his choices allow him to develop wisdom about the meaning of life.
This reflects the theme by underscoring how a simple choice can have significant impact on all of one’s future choices, and while the destination of life may be the same for all (death), each person’s journey is unique.

The traveler finds himself at a fork in the road, a spot in the woods where "Two roads diverge" (line 1). He is, apparently, walking through the forest when he reaches this spot, and he spends some time considering which of the two roads he would like to take. He peers down one until it bends away into the bushes. He peers down the other, noting how grassy it is. He compares them, seeing that "the passing there / Had worn them really about the same" (9-10). In other words, then, about the same number of people have taken each path, as they are equally worn by the footsteps of other travelers. This traveler is torn about which road to take—this is the problem he faces—but he eventually decides on the second, saying, "Oh, I kept the first for another day!" (13). However, he realizes that it is likely that he will never be back in this exact same spot.