How does a DC motor drive control the motor?
It doesn't drive the motor.  I assume you're talking about a brushed DC motor.  Brushed DC motors have the commutation built into the mechanics of the motor - the brushes are part of that.  You can just apply a voltage and it'll go.  But since voltage relate to speed and current relates to torque then you can start doing fancy things to control those variables to have the motor behave in a desired way.  Brushed DC motor controllers are there simply to help with one or both of those parameters but are not involved in commutating it.