It's not just in Mexican society, but in many cultures around the world, the third birthday is so important because if the baby lives to age 3, there is a good chance it will survive to adulthood (they are not adults at age three -- ignore that ignorant information). In the previous hundreds of years, lots of babies died in their first year. So if the baby turns three, it is a good sign that the baby is strong enough to survive. That is a big deal, especially if previous babies in the family have died.
are you talking about the Qunceñera?...if so, then according to Mexican culture, that is when a girl becomes a woman...it's like a right of passage. but if your talking about the third birthday then I'm not sure.