The speaker compares his heart to an empty nest in order to emphasize the fact that his love has left.
The nest is a metaphor for his heart. When he is in love, he is so happy, that he can't contain his joy. This is referenced when the speaker states, "I cannot still it." He then relates this to chirping birds in a nest. However, when his love is gone, the nest becomes empty and depressing. The speaker is also feeling empty and sad because his love is gone.