It’s like Will never learned how most other people sit down on chairs, but everyone in the future is so respectful of cultural differences and quirks like that, no one ever said anything.
Yeah, I have always wondered what was up with that. Did Frakes sit down with himself and say: How do I make this Riker character more standouty? I know, I’ll have him throw his leg over the back of chair to sit down!
Note that as the show went on and a certain person’s uniform kept getting tighter around the middle this happened way less often. I’m just sayin’.
I’d guess that it’s an acting thing; it lets Frakes enter the scene and sit down quickly while keeping his face in the frame. Look at the last example, where Riker and some other person enter and sit down together. Riker is seated, balanced, looking up, and already speaking before the other person has finished sitting down. (Then he can make that dramatic odalisque pose.) In many of the scenes, the camera is focused on Riker as he is sitting, often when he’s speaking. The Riker maneuver means he doesn’t have to turn himself or the chair, or bend awkwardly, which might disrupt the shot. These all seem minor, but they can be very important for a director on a (time and money) budget. It took so long for many of us to notice the Riker maneuver (the Picard maneuver was much more noticeable, I think) because the directors are usually making the most of what it provided.
Also, look at some of those chairs. Some of them, particularly the Ten Forward ones, look pretty small, and might not swivel. How would you sit down on them? Remember, you don’t have time to adjust yourself when you do. Also, you can’t look down.
Finally, Jonathan Frakes is 6’4". This is just how you deal with chairs when you’re tall.
I read on Reddit that Frakes actually had some sort of back injury from a furniture moving job he’d had in the past, and this was a way of alleviating symptoms from that when he had to do take after take of coming into a room and sitting down. It also explains the “Riker Lean” when he’d be leaning on various things in his shots.