What´s the logic behind the Power Curve?
why does power affect airplane´s aerodynamics?

It doesn't. The aerodynamics will determine how much excess power is left for acceleration or climb. But not the other way around.

Why is it important how much power […] the engine has?

Because the power of a piston engine is constant for the same air density, regardless of speed. It is more helpful to give a power rating because that figure is valid over the whole speed range, unlike thrust. Thrust is inversely proportional to speed and so a thrust figure is only valid for exactly one operating point.
Pure turbojets have approximately constant thrust at subsonic speed, so here the thrust rating is more useful. The power a turbojet can produce increases with flight speed.