Why is adiabatic process isentropic?

By definition a reversible adiabatic system has $dQ = 0$.

We also know the following from the Clausius Theorem :

$dS = \frac{dQ}{T}$

Then it is easy to see that there can be no change in entropy.

Note that irreversible adiabatic systems CAN see a change in entropy because in that case the above equation is no longer an equality but an inequality :

$dS < \frac{dQ}{T}$

Consider an isentropic (idealistic adiabatic process, with no friction) process, in which the volume of a system is increased. For that to happen, the internal energy of the system would decrease because the system would be performing work. So any gain in the number of available microstates from an increase in volume is negated by the loss of internal energy by the system, which decreases the number of microstates (lower temperature means smaller molecular speeds means fewer microstates).

Similarly, isentropically decreasing the volume of a system would seem to decrease the number of microstates, but work must be done on the system for this change to occur, causing an increase in the internal energy of the system, which counteracts the decrease in volume and keeps the number of microstates the same.