Why does a body rotate faster if force is applied away from the pivot? I need an intuitive answer to this. Like why does a door rotate slower if I push at it closer to its hinge, or faster if I push at it farther away from its hinge? What actually makes that happen?
You can use the formula
$work = force \times distance$
to imagine it intuitively.
When the pushing force is close to the pivot, the force moves through a small distance (for a given angle of rotation), the work done is low, so the kinetic energy gained by the door, related to the speed, is low.
When pushing with the same force further from the pivot, the distance is larger, for the same angle, so the kinetic energy gained and speed of the door would be larger.
Imagine a 100 metres wide door. It'll take a loooong time to close this door via pushing at the edge because you'll have to run ~300 metres. Your force only moves the door a small but for each second, which is why it is easier to do. But also slower.
If the door is not very heavy (just styrofoam e.g.), you might be able to close the door much faster by pushing further in, because that can speed up the edge faster than what I could run.
But pushing at the hinge will also not work, since then you aren't causing any rotation at all.
The optimum must be somewhere in between.
If we consider torque alone, then the created torque increases the further out you push (where the lever length is longer):
So this indicates that it should be easier and faster*, the further out you push. But don't forget that this requires you to be able to continuously apply a constant force $F$.
If that is possible, then naturally the object will rotate faster when pushing further out. If this is not possible as in my example above due to geometric reasons, then further out is not necessarily better.
* Faster, because a larger torque causes a larger angular acceleration which from a stationary initial state causes a faster rotation.