Why does the control effort increase if the bandwidth of the system is higher then the bandwidth of the plant?
Intuitively (and I leave it as an exercise for you to work it out with math), it's because you have to push harder on the plant to get it to move fast.

Consider a bowl of honey and a spoon -- moving the spoon around in the bowl slowly takes little effort, but moving it fast requires a lot of effort, because honey is viscous.

It may be most informative to take a simple plant model, like a 1st-order low-pass filter, and without ever designing a closed-loop controller, design a signal for it to follow that has significant energy content above it's rolloff frequency.  Then look at the relative amplitudes of the components of that drive signal compared to the resulting components of the motion of the plant.

Even in open loop your control effort is higher at higher frequencies, for reasons very much analogous to that spoon in honey.