Propellers typically rotate clockwise when viewed by the pilot inside the cockpit, and the air pushed backwards by the propeller also rotates. This corkscrew pattern of airflow moves backwards over the aircraft and strikes the tail from the left, pushing the tail right and yawing the aircraft to the left. This is known as the slipstream effect.
Any increase in thrust will increase the yaw from the slipstream and any decrease in thrust reduces it.
Propeller torque is a result of one of Newton’s Laws of Motion: for every action, there is an equal and opposite reaction. The clockwise rotation of the propeller has an equal and opposite effect which tries to rotate the aircraft anti-clockwise.
During takeoff, this puts slightly more weight on the left wheel – increasing friction on the left side and leading to yaw to the left.
When the aircraft (and propeller) are at an increased angle to the relative airflow (i.e. an increased angle of attack of the wing), the down-going propeller blade moves through a greater distance than the up-going blade. This is because the downgoing blade starts slightly rearwards when pointing vertically up and finishes slightly forwards when pointing vertically downwards.
This means the downgoing blade has a higher angle of attack and thus creates more thrust when compared to the up-going blade.
Since the downgoing blade is on the right, this results in yaw to the left.
This is known as the Asymmetric Effect.
Gyroscopic effects are talked about in more detail in the Aircraft General Knowledge course & exam. For now, all you need to know is that when a force is applied to a rotating object (such as the propeller), gyroscopic precession rotates that force 90° in the direction of rotation.
In the case of propellers, when the aircraft is pitched nose down it is the equivalent of applying a force to the top of the propeller disc. Gyroscopic precession then rotates this force 90° in the direction of rotation (clockwise).
So the force at the top of the propeller is rotated such that it is applied to the right-hand side of the propeller (view from the cockpit), creating yaw to the left.