Moon has no atmosphere because

The moon's escape velocity is much less than the root mean square (RMS) velocity of gas molecules on the moon. Therefore, the molecules of gas escape from the moon's surface. The gravitational field is weak on the surface of the moon. 

As a result, the moon lacks the capability to hold an atmosphere, leaving it exposed to the vacuum of space. This absence of an atmosphere also means there is no air to moderate temperatures or shield the surface from cosmic radiation. Hence, the moon doesn't have an atmosphere.