If there were no friction, then the spindle would simply be dragged forward due to the horizontal component of the force F. The force would also induce a torque of magnitude Fr on the spindle and cause it to spin in the counter-clockwise direction. Friction between the floor and the spindle however, opposes this with a force f acting in opposition to the force F. This friction force acts on the wider circular part of the spindle and induces a torque fR in the clockwise direction. This frictional torque is eventually responsible for making the cylinder roll without slipping on the floor.
The force of gravity passes through the axis of rotation and does not induce any torque on the system. Let the angular acceleration of the cylinder be b. As mentioned there are two torques acting on the spindle, i) due to the friction of magnitude fR in the clockwise direction and ii) due to the force F of magnitide Fr in the counter-clockwise direction. So we have,