More voltage means more torque at the same speed.
A useful first-order model of a motor is a resistor in series with a voltage source. That voltage source is a function of the motor speed, and apposes the applied voltage. Torque is proportional to the current.
This simple model tells you a lot. When the speed is 0, the current is just the applied voltage divided by the resistance. That's the stall current, and is when you get maximum torque.
If you let the motor go (no load on it), then it will accelerate due to the torque. That makes it go faster, so now the internal voltage source starts offsetting some of the applied voltage. Eventually the motor will get to the steady state speed for that voltage. That is also called the unloaded speed of the motor at that applied voltage. This is where the internal voltage source has offset most of the applied voltage. The little that's left across the resistance allows just enough current to flow to produce just enough torque to overcome the friction and other losses to keep the motor spinning.
Now you should hopefully be able to understand why the first sentence is true.
One gotcha of this is higher power dissipation. The motor's dissipation comes from the resistance, and is therefore proportional to the square of the current.