If you chance upon a bird's nest, in any tree or on the ground, with fledglings or eggs, and the mother is sitting over the fledglings or on the eggs, do not take the mother together with her young. Let the mother go and take only the young... (Deuteronomy 22:6)
Nachmanides writes that, on the most basic level, the reason for this mitzvah is to teach us compassion.
Taking the child within the sight of the mother would cause the mother acute pain. A mother's love and compassion for her offspring is, in Maimonides' words, "not a function of the intellect or speech, but a function of the thought process that exists in animals as well as in people."
By performing this mitzvah, we are training ourselves to feel empathy for all G-d's creations.
According to the Zohar, this commandment also has a profound cosmic impact by arousing heavenly mercy for the Jewish people.
When the mother bird is driven from her nest, she cries bitterly and despairingly over her separation from her young.