A dungeon master, plays a very different game from the game that the players are playing.
A dungeon master, or game master, or story teller, is playing a game of tag while players are playing whatever game they want to play, with each other if they understand D&D.
Whatever game system you are playing, the point as the centre point of the players trust is to deliver a fun experience. That may seem gross and derivative but the point is true.
The same way we (the biblical we) watch action, horror, suspense, and mystery movies, among others, or read adventures, mysteries, ..or y'know, consume art, period. We look for emotional response of ourselves in consuming artistic material that others have created. Music, imagery, literary, whatever.
D&D is no different thereby but II will say it gets a lot more intense the more the consumer explores what it provides.
Discovering yourself in the game.
Many people represent themselves in D&D whether they mean to or not. I have some acting ability but not much to speak of. I have some voice acting ability, but not much to speak of.
What I have is enough panache to carry some specific characters, and deliver some specific story beats.