Oggers, I'm reposting your Otto cycle diagram from your interesting post #118.
Work accomplished can be seen in the diagram. Energy added by burning the air/fuel charge and the rapid pressure rise is represented by line 2-3. Residual heat remaining when the exhaust valve opens at point 4, is represented by line 4-1. Notice carefully that line 2-3 is longer than line 4-1. The relationship between heat and pressure in a gas is well known, therefore heat has been converted into mechanical work between points 3 and 4, to yield a shorter pressure line 4-1 than pressure line 2-3.
Efficiency could therefore be expressed as the ratio between the heat added to the process to raise the pressure from point 2 to point 3 (a larger pressure change), and the residual heating remaining in the pressure drop from point 4 to point 1.
Why then does higher compression yield more efficiency? Work is done by the pressure difference between the top and bottom of the piston. Higher compression increases the pressure difference, and therefore converts more heat in the pressurized gas into mechanical work, yielding a residual heat value that is lower, and an efficiency ratio that is higher. Notice too that most of the work is done in the top half of the expansion (or power) stroke, where most of the benefit from higher compression is located and is most useful.
In the turbine and rocket engine, higher pressures yield higher velocities imparted to the reaction mass, and thrust as we know is a product of the velocity and mass. The higher the velocity imparted to the same mass, the higher the thrust. However is a turbine, lower velocities imparted to a larger air mass (the turbofan) is more efficient that adding higher velocities to a lower air mass (pure turbojet). With either turbine type, higher compression (or pressure ratios) yields higher velocities.