How to understand these verses from the poem "The road not taken" by Robert Frost

Solution 1:

In leaves no step had trodden black.

This line refers to leaves that have been stepped on to the point where they are black instead of the color they were when they fell off the tree.

The word "tread" means (one of its meanings) "to step on." So you can tread on the leaves as you walk along. If you have stepped on the leaf, we can say you have trodden on it.

So the speaker is saying (at this point) that that road has not been walked on. The leaves have not been stepped on so they have not turned black from being crushed or ground into the dirt or anything.

And that has made all the difference.

In this case, the narrator, reflects that, of the two paths he could have taken in life, he chose to follow "the one less traveled by," and he believes "that has made all the difference." Basically, the narrator reached a point in his life when he was faced with decision to either conform to society and live a "normal" life, or to live his life in way that embraces individuality or an less conventional existence than the first path.

The road that fewer people had taken ("was more grassy and wanted wear") was more inviting to the narrator, so he made the choice to follow the path that fewer people had taken. The narrator believes that chosing to follow the norm would have been the wrong decision.

Source

Solution 2:

To expand on Kiran's answer, the reason many people give that interpretation of the poem is these lines:

Though as for that the passing there
Had worn them really about the same,

And both that morning equally lay
In leaves no step had trodden black.

That is, he picked the one which was 'least chosen', but the difference between them was actually small or illusory. Both had been worn 'about the same' and were 'equally' covered in leaves so regardless of which path the speaker chose, the last stanza would still apply:

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.

The routes are different, but the decision was arbitrary. Both would, presumably, have resulted in interesting and unique life experiences - one wasn't more valid than the other, and in both cases, the speaker would experience regret at missing out on the experiences of the other path.

This feeling is pretty universal, which I think is one reason the poem continues to be so popular.