The law has traditionally been explained as sparing the mother the painful sight of seeing her offspring taken away. However, it is not likely that chasing the mother away would spare her the pain, since forcible separation from her young and finding them gone later would also be painful. …What the text finds callous are the acts themselves, quite apart from any impact they may have on the mother.
Tigay Jeffrey JPS Torah commentary
Since therefore the desire of procuring good food necessitates the killing of animals, the Law enjoins that this should be done as painlessly as possible. … It is also prohibited to kill an animal with its young on the same day to prevent people from killing the two together in such a manner that the young is slain in the sight of the mother; for the suffering of animals under such circumstances is very great… and does not differ from that of man, since the love and tenderness of the mother for her young ones is not produced by reasoning but by imagination, and this faculty exists not only in man but in most living beings…The same applies to the sending away of the mother bird. The eggs which the bird sits on and the young that are in need of their mother are generally unfit for food, and when the mother is sent away she does not see the taking of her young ones, and does not feel pain. ..
(Maimonides, Guide for the Perplexed, III:48)
The ruling of the mother bird is not based on the Almighty’s pity for the animal. Otherwise He would have forbidden us their slaughter. The reason however for the prohibition is to teach us compassion and the avoidance of cruelty. Butchers and slaughterers become hardened to suffering by their occupation. These precepts of not slaughtering the mother and the young on the same day and sending away the mother bird are not inspired by feelings of consideration for their suffering but are decrees to inculcate humanity in us. In the same way our Sages regarded all the Torah’s precepts, negative and positive, as decrees.
( Nahmanides, Commentary on Devarim 22:6)