The speed of a motor is determined by the voltage and the torque by the current. If a motor is running at a certain speed with a constant torque and the load increases, the current will increase and so also the torque to maintain the same speed.
To be a little more precise there is a small voltage loss in the motor that does not contribute to the speed. This is called the IR drop. The current through the motor times the resistance of the motor creates a small voltage that must be subtracted from the applied voltage. As the load and current increase, this IR drop also increases causing the motor to lose a little speed.