Frost is probably the only twentieth-century poet whose work has actually entered the collective American consciousness. This can be seen in the fact that even people who are not English majors or literary specialists can often quote his work extemporaneously—or are at least very familiar with it.
It's partly the arresting nature of his wording, as in lines like "Something there is that doesn't love a wall" and "Warren? she questioned / Dead, was all he answered," that accounts for this. Additionally, the fact that his verse grows out of ordinary situations but has universal meaning is what makes it understandable and relevant. The best example is probably "Stopping by Woods on a Snowy Evening."
Two anecdotal facts are also worth mentioning.
First, though I was very young at the time, one of my earliest memories is of Frost reading his poetry at John F. Kennedy's inauguration in January, 1961. In the freezing cold of that day, Frost presented a heroic image, in spite of (or perhaps, conversely, because of) the difficulty he had reading his own words, after which he recited from memory.
Second, the use, in a television commercial about cancer treatment, of a few of his most famous lines—
But I have promises to keep,
And miles to go before I sleep,
And miles to go before I sleep
—is incredibly moving and poignant. There are few poets from any time—apart from the very greatest like Shakespeare, Shelley, and Tennyson, for instance—whose work, when quoted, has a similar power.