When interpreting a work of art, a person's judgments about the work of art should be based on evidence from the work of art itself. That still often leaves room for interpretation, but it should stop a person from believing just whatever one wants to believe about a work of art.
The speaker says in this poem that the roads were basically the same. There is no question that he did or didn't take the road less traveled--he did not. The speaker says he did not. They were the same.
The sigh can maybe be interpreted in different ways, except that the sigh and the last stanza cannot be interpreted as if the speaker is happy because he took the unpopular, less traveled path. That idea is simply not present in the poem.
I suggest the sigh is similar to a chuckle. The speaker is preparing to tell a white lie or a big fish type of story. He is going to play the wise old man and tell a story about how he chose the unpopular path, and say that that has made all the difference. This is a tale, though. In actuality, he does not do that. He may even wish that had been the case, but it wasn't.
I shall be telling this with a sigh/Somewhere ages and ages hence:/...I took the one less traveled by,/And that has made all the difference
How many times have we heard people wonder what their lives would have been like if they had chosen a different spouse, a different career, taken advantage of a different opportunity? The speaker sighs, indicating, perhaps, some rue about his choices in life; he talks about his choice for years--these actions seem to indicate some regret, some wonder about "What if I had it to do all over again?"
Indeed, in Frost's "The Road Not Taken,"these are the reflections of an old man who evaluates his life. It is natural for people to wonder how their lives would have been different if they made a different choice. After all, there is a little of Hamlet's ambivalence about life in everyone.
In the poem "The Road Not Taken" by Robert Frost, I believe the sigh refers not to the fact that the two raods were more or less the same, but to the fact that he chose the one less travelled by. The poet seems to be pleased, in the end, at his choice of road, but I am wondering if because it was less travelled, it was a lonely road? the poems of Robert Frost are enigmatic and that is what makes them so interesting - because people puzzle and argue over them - as we are doing now! However, the really interesting thing is that Robert Frost blames us for it! He once said that people tried to read too much into his poems - yet he chose every word so carefully. I think he is sighing over the fact that actually in the end,the choice of road didnt matter as much as he thought it would - in hindsight.
In my opinion, when the speaker sighs, it will be nostalgic, but it will be a sigh of fake nostalgia -- the speaker will be fooling himself.
In this poem, the road the speaker is taking is no different from the other road, the one he didn't take. He says that they are both equally worn, one as good as the other. But at the end, he claims that the one he took will make all the difference. That seems unlikely since the roads were not really different.
So he's sighing out of nostalgia, but he's really fooling himself.
The sigh imagined in the last stanza is not to be taken as an expression of regret for a life wasted, but as a semi-comic picture of the speaker envisioning himself as an old man, wondering how things would have turned out if he had made a different choice—which is not at all to imply a rejection of the choice he did make.
If you remember from the poem, the roads were almost exactly alike in three ways. His sigh is definitely not reminiscing, but comical, like choosing french toast, and thinking back if you had chosen pancakes. Would it have mattered? No, that is what is so amusing.
I think the sigh is that of remembering; the narrator chose the path that was difficult yet rewarding, and the memory brings back certain emotions and feelings that makes one 'sigh' upon recalling.