Ultimately it comes down to the first law of thermodynamics: energy is conserved.
If you start in straight and level flight and then increase the throttle setting, the engine puts out more energy every second, so where can that energy go?
IF you pitch up to maintain speed, you will also maintain power lost though drag, so that extra power can only escape as a positive climb rate. Nowhere else to go.
IF you pitch to maintain height, then the only place the energy can be dissipated is through higher drag. A higher speed will achieve this, (unless the wings come off). Theoretically, a much lower speed also works, but fear of the stall makes that less popular.
Practically speaking, if you want to change speed quickly, nothing beats gravity: its force is 5-10 times greater than your typical spam-can engine's maximum thrust.