My big "DM insight" that I haven't seen anywhere else is that engagement has two parts,
an emotional component and a sensory component.
The emotional component is aided by a game where it's clear how it's set up, how it works, what the mechanics are and how the DM's decisions are made. You can care about your character and the game world, let it be "real" because some of the processes are so externalized away from any one person's decisions.
So as a DM, I use a transparent, almost hypermedialized method for this. Rolling openly, making it clear what rolls mean what, sometimes showing things from the book, revealing some of the numbers from the encounter tables. This is basic for many story games that have very visible and easily engaged mechanics.
The sensory component is aided by a game where the mechanics are invisible, where all you you have to do is close your eyes and dream away. You can imagine the smells, sounds, weights, sights of the fantasy world. These two components are often but not always in conflict.
Whenever I add mechanics I work hard to make them invisible, or DM-facing only, and when I get the vibe that the players need more buy-in, when they think that I'm just bullshitting, I reveal some of the mechanics. The mechanics should be solid, the players can rely on the mechanics creating and driving the events in their game world. This is to take away the sense of unease, to lessen the perceived power disparity between player and DM.
Mechanics should also be associated directly with the goings on in the game. The cart should not be pulling the horse.
Games often (there are exceptions) thrive on choice and a solid mechanical landscape is great place to make solid and consequential choices.
And if you're always always always being made aware of the mechanics, it's taking away your dreaming power.
Here's an analogy. In early TV news, they would be very visible with their documents and reports in hand as paper, and cameras and mics would be visible. This makes it so that the audience gets a feeling of what's going on, "this is real". It builds buy-in.
And when you watch a movie, you want the frame to fall away so that you just focus on the goings on in the film, you don't want to see any cameras or mics in Middle-Earth (although... wouldn't that be awesome?). It builds presence.
Anyway... it's sometimes a tradeoff, sometimes a balancing act, and sometimes something you can do without compromise because you can find mechanics that are both solid and non-intrusive.