For me, bouncing around while maintaining a high cadence was the result of two things - my technique and my inexperience. For years I pushed a big gear but as I got older I noticed others spinning more quickly and appearing to be "fresher" after a long ride. The first thing I focused on was saddle height and setback. This is tricky and took some experimenting and patience. Still fidget with this at times. The other was a fluid motion as spinning quickly is different than "mashing" a big gear.
As others have commented, position is important in terms of where your a$$ and pelvis is situated. Working on thrusting forward, leaning over, sitting upright, etc, while still maintaining a high cadence all have to do with technique and putting in the training. After a while, I found that "spinning" was more efficient than "mashing". It just took practice.