What is the significance of the sigh in the last stanza?

The sigh near the end of "The Road Not Taken" is intended to express a feeling of regret that the speaker will never know what would have happened to him in life if he had taken that other road, the road which he calls the one not taken. Obviously, if he had taken the other road, then the road he actually took would have been the one not taken. This poem is obviously not about a man taking a walk and having to choose between two real roads. The crossroads in the poem are a metaphor for a time in the speaker's life when he had to make an important decision about how he was going to live his life. Some have speculated that the speaker of the poem is Robert Frost himself, as he had to make a decision about a very common life problem. He knew he had creative talent, but he still had to make a living. He could either have a spartan existence, not unlike that of Henry David Thoreau, and devote his life to poetry; or he could get some kind of job and only write poetry in his limited free time. Frost chose the former "road." It was a long, bumpy road, but he was one of the few poets who achieved recognition and financial security. If he had taken the other road he might have been equally successful without having to endure the rather dreary life of a New England farmer. But he would never know.

Frost confided to a young girl who queried about the meaning of the "sigh":

Amherst Mass April 1925

"Dear Miss Yates:

No wonder you were a little puzzled over the end of my Road Not Taken. It was my rather private jest at the expense of those who might think I would yet live to be sorry for the way I had taken in life. I suppose I was gently teasing them. I'm not really a very regretful person, but for your solicitousness on my behalf I'm
your friend always
Robert Frost"

(Finger, L. L.: "Frost's 'The Road Not Taken': a 1925 Letter come to Light", American Literature v.50)

The "sigh" is one of the many subtle amenities in "The Road Not Taken." It occurs in these lines:

I shall be telling this with a sigh
Somewhere ages and ages hence:
Two roads diverged in a wood, and I--
I took the one less traveled by,
And that has made all the difference.

We seem to be hearing that sigh across those "ages and ages" at the point where the word "I" occurs twice with a dash indicating where the speaker pauses and takes in a deep breath before continuing.