By age three, a child usually understands enough to begin learning about the commandments. The first haircut is a way to teach the child about the prohibition against too closely cropping the hair on the sides of the head. The Torah is the Tree of Life. At three, when the child's understanding has developed, then his parents can start teaching him the Torah and he can start doing some of its commandments. He finally gets to taste the sweet fruits from the Tree of Life.
Some people honor the first haircut with a festive celebration.
They express thanks to G-d for allowing them to teach Torah to their child. Many are accustomed to take their child to a great Torah scholar who cuts the first snip. Not everyone has this custom, however. I found this out when my son was about to turn three, and I approached a renowned Jerusalem rabbi, asking if he would like to take the first snip. Some people weigh the cut hair and give that weight in gold or silver to charity, especially a charity which promotes Torah study.