Pull up and the airplane turns left, push down and the aircraft turns right. Its called gyroscopic precession. You probably studied it in 10 grade physics.

Another force known to turn the aircraft is called P-factor. If the aircraft is travelling through the air mass at a positive angle of attack, then the prop is also. In this case the decending blade's pitch is greater than the accending blades pitch. The decending blade is on the right side of the engine and thus turns the plane left.

There is also a third force working here. Torgue. The engine is spinning one way. The airplane wants to spin the other way, left.

Fly any real airplane and you will learn that you must retrim the airplane when you change speeds. If the airplane has a very large speed envelope, then there might be a good deal of trim changes as you speed up or slow down.
The airfloil also plays a role. Flat bottom airfoils require much more tirm changes as speed is changed. Symmetrical foils require less trim changes throughout their envelope. High wing aircraft are much, much more susceptible to large trim changes with throttle changes than are low wing aircraft.

Down thrust helps reduce the trim changes throught the speed envelope . Thats why we use it.