What I wrote above is what the hip does once it's set in position for the swing. I should add setting the hip in the correct position consistently is the difficult part and is the biggest factor in differentiating the levels of tennis skill. This is commonly referred to as the footwork, the main purpose of which is to set the hip in the best spatial position and orientation at the right timing.
For this skill, either by instinctive or learned coordination, one should know precisely where the best contact point is in relation to the hip and have well developed and conditioned (strong and flexible not muscular and bulky) legs muscles all the way from the toes to pelvis to place the hip there quickly and accurately.
If you want to improve the hip usage, first stretch the hell out all the muscles around the pelvis (which usually are terribly tight and shortened) and work on balances on each foot (even start by just standing on one foot and see how good they are).
One of the most important part of tennis skill won't come easy however and will require a significant commitment for improvement (lots of time and effort) to see good results.