In this week's Parsha (22:28), we are told not to slaughter an animal and its offspring on the same day. This prohibition applies not only to animals that are to be brought as offerings but also to other animals that will be eaten. It is a matter of discussion as to whether or not this commandment applies to the father as well as the mother. Why shouldn't “it and its young” be slaughtered on one day? How does adhering to this precept make one a better person?
The consensus of the Meforshim seems to be that this is one of many commandments meant to teach us sensitivity and compassion. Other Mitzvot in this category are not cooking a kid in its mother's milk and chasing away a mother bird before disturbing her nest. How do these commandments teach us sensitivity and compassion? They all deal with the relationship between a mother and her offspring, the closest relationship that exists in the animal world. (This is why, despite the use of the masculine noun, אותו, it is debatable whether the Mitzva in this week's Parsha applies to a father or not.)
Rabbi Shimshon Rafael Hirsch says that the essence of the animal is the selfishness that pervades all of its relationships; the exception to this is the relationship between a mother animal and her young. In this case, the mother exhibits selflessness and a willingness to sacrifice for others. This care displayed by a female animal for her young is the closest an animal ever comes to humanity. Thus, when we are about to eat an animal, to make that animal a part of us, we make sure to keep in mind the kindness inherent in that animal by not killing the mother and the child on the same day. By remembering that such a relationship can exist between animals, we remind ourselves of our capacity for selflessness and willingness to help and protect others. If we are commanded to be so careful about our treatment of animals that one is about to slaughter, how much more so should we be sensitive in our treatment of others?
Along the same lines, there are many Halachot in addition to those mentioned above that provide for the kind treatment of animals. Their purpose is to teach us compassion. Hashem, in order to guide us away from cruelty and teach us kindness, has given us many laws regarding the proper treatment of animals. Through these laws we learn to be sensitive to the needs of animals and treat them with compassion and kindness. If we have to behave this way toward animals, we must behave this way toward each other. In addition, through acting in a kind and compassionate manner toward animals, we strengthen our Midot of kindness and compassion, thereby making those Midot a greater part of our nature and making it easier for us to behave in an appropriate manner toward our fellow human beings. So, not killing an animal and its child on the same day really does make one a better person.