What does "stick the hydraulics under something" mean?

He is using jargon to state that they use a metaphorical hydraulic mechanism to force more profit through the system.

(Hydraulics, such as the lift mechanism on an elevator, or the brake master cylinder in your car, is a method of using a liquid force enhancer to move objects).

And, they “stick it under something” because it is a complicated mechanism and not completely ethical and they don’t want people looking at it too closely.

In the context of the video, where the “interviewee” is explaining to the “interviewer” (and I hope that you fully comprehend that this is a satire / parody video, not a real explanation of anything… no matter how close to reality it is) the reason why some fees are “jacked up” (hence the reference to “hydraulics”, in the sense of a hydraulic jack).

Hide the details, sell the end result.