I do not understand what power actually is
You can think of what happens to electron flow in the wire. Although not true, try to think of the electron as a mechanical particle. Whenever it tries to move whithin the wire, it hits something and that collision generates heat. So you can think of a energy being transfered from kinetic energy of the electron to heat (so electron speed falls down in that moment). So the electrons do not have constant speed all the time altough we may say they have an average speed and this average speed depends on the resistance of the wire which is exaclty this obstacles that the electron hits.

The wire would not became hot if it does not have any resistance. So no power would be consumed by the wire.

When you double voltage and also double the resistance, you may think that the electric field within the wire is higher so the electron may reach a high speed faster than with lower voltage. But also the resistance is higher so it may hit its obstacles in a stronger way. So the average speed may became the same (current is the same) altough now you are dissipating more heat because collisions are stronger.

This is a very gross way to think but it may help you imagine why things are the way they are by some analogies.

Also, you may think power as Joules per second (Watt). So it relates some energy unit per time. In ohms law example, it applies to heat dissipation. In other words, how much energy is being wasted in heat throughout the wire. If you think of mechanical systems, power may represent how much energy is necessary to move something (you can calculate it's minimum kinetic energy to achieve a desired speed and so calculate how much energy you should transfer to this object to reach that speed). So, as power is direclty related to energy, you may think that energy is always being transfered from one manner to another. Power may indicate how fast this things happen.