This relatively short poem by Matthew Arnold focuses on the concept of unrequited love and how this contributes to the state of isolation in which all human beings ultimately find themselves, whether they completely realize this or not. According to Arnold, the only real difference between happy people and unhappy ones is that the happy people are laboring under the misapprehension that they are actually fully bound to another person, whereas unhappy people, such as the speaker, have been forced to recognize that this is not true. In fact, we can never trust that somebody else's thoughts and feelings are aligned with our own, and once we have had this fact revealed to us, we can never again become ignorant of it and return to the state of happy ignorance in which we once existed.

Arnold explains that these "happier men" are those who:

Have dream'd two human hearts might blend
In one, and were through faith released
From isolation without end

However, he is clear in his statement that these people are not actually any less alone than the speaker's own poor heart, whom he addresses as "thou" throughout the second part of the poem. Having shifted his focus from Marguerite, to whom he now bids farewell, to the personification of his own heart, the speaker seems to be trying to console his wounded heart by stressing that it is not alone in being alone. Rather, those who are happier are simply existing in a state of "faith" that they are not isolated, while the isolation remains the same.

Yet she, chaste queen, had never proved
How vain a thing is mortal love

The above quote is one of the key concerns the speaker has. He describes how Luna, the goddess of the moon, was never able quite to recognize this because she was not human; however, others who are human do come to realize the truth of this axiom because they want so much for it not to be true. In binding themselves to other people and committing their hearts to love others, they are trying to escape the inescapable fact: 

Thou hast been, shalt be, art, alone.

For the speaker, it does not seem to be bidding goodbye to Marguerite that is the most painful aspect of what he has experienced. Rather, he is bidding goodbye to the faith which allowed him to chase happiness in his life. Now, because of the betrayal he has experienced in life, there is a suggestion that he will not be able to commit himself again to loving another person. He has been disabused of the notion that two hearts can ever truly beat as one, and this has left him sadly aware of the true nature of existence.