It is simplest to view the torque as consisting of two pieces:

One part that has the same direction as the object's rotation axis, and

One part that is at a right angle to the rotation axis.

The portion of the torque that lines up with the object's rotation axis will change its angular speed, just the
way that a linear force acting along the direction that an object is sliding will change its linear speed - it
will speed up if they act in the same direction, or slow down if they are in opposite directions.

The part of the torque that acts at a right angle to the spin direction will not make the object's rotation speed
up or slow down, however. It will change the spin direction - a phenomenon known as precession.