Why is the lunar month longer than the sidereal month?

Because in 1 lunar month, the earth travels 1/12 of the way around the orbit around the sun, so to get from one lunar phase to the next one a sidereal month later, you have to add on a little bit of time to account for the changing geometry of the earth-moon-sun system. This makes the synodic lunar month a little over 2 days longer than the sidereal month. The later measures the time interval between the moon being in front of the same background stars.