There are two issues at play here.  The first would be the meaning of the quote.  The second would be explaining how Frost's poem addresses the quote.  In terms of the quote's meaning, I think that its idea is that the ride is a part of the journey that is life.  The "rides" we take help to define our being in the world.  Life is constructed by the various choices and rides we take in life.  Our being is the sum total of our choices.  While life itself is larger than one ride, the various rides we take and the attachment of choices to each ride help to form what life is and what it can be.  Accordingly, Frost's poem speaks to such a condition in a clear manner.  Frost speaks of "promises to keep" in the last part of the poem.  While the speaker might want to see what lies in the woods and what beauty might exist there, social obligations and personal bonds of responsibility prevent him from doing so.  It is here where one sees how the speaker's life is defined by the ride taken.  The speaker's ride is one in which the obligations made.  The ride in which "promises" were made and must be kept is what defines his being.  Life, for the speaker, comprises of this ride.  The speaker's ride in which these promises were made might be over, but life is defined by them.  It is here where I think that Frost's poem speaks to the idea that while the ride might be over, the implications of the ride ("promises to keep") is where life resides, defining the being of the individual.  In this, Frost's poem speaks to a life in which rides may end, but life still continues as a consequence of the rides taken.