Why do only female mosquitoes bite?
It comes down to reproduction. Female mosquitoes need the proteins and iron found in blood to develop their eggs. Without a blood meal, they can't produce viable eggs. This biological need is what drives their biting behavior.
After feeding, a female mosquito will rest for a few days while her eggs develop. She can then lay anywhere from 30 to 300 eggs at a time, depending on the species. Once she's laid her eggs, she'll seek another blood meal to start the cycle again.
This is also why signs of mosquitoes around your yard, like standing water and increased biting activity at dusk, are worth taking seriously. Where there are biting females, there are eggs, and where there are eggs, there's a growing population.