I feel I must begin by saying that I don't believe in wrongbadfun. That is, you and I may not get the same thing out of the gaming experience, but that's okay. Now, it's possible that you and I don't have compatible play styles and that, to minimize frustration on both sides, we maybe shouldn't sit down at the same table, but the following is not meant to be a manifesto on how or why everybody ought to play, only to give a sense of how and why I play. You do you, bruh.
Because I am in the humanities/social sciences, I'll begin with a definition. To me, tabletop roleplaying games (TRPGs) are cooperative storytelling augmented by improvisational acting. That can mean a lot of things, though, so I'll expound a bit.
Cooperative storytelling in this sense means that, although each player brings a separate character (PC) to the table, they are striving to braid those disparate stories together to create a larger whole with meaningful connections between the PCs. This does not mean that you can't have your revenge storyline; it just means that at some point I should have a reason to care about it too. Inigo has a very personal story, but Wesley has a stake in it, both because the Six-Fingered Man is a minion of the evil Prince and because by the end Inigo and Wesley are friends. From a TRPG perspective, it would be really easy for Inigo to keep his revenge story to himself and pursue it alone. But I prefer games where Wesley gets to say "Someone was looking for you" to Count Rugen.
Improvisational acting in this case simply means that players speak in first person and announce actions with first person pronouns rather than third. (i.e. "I open the door" as opposed to "She/my character opens the door".) I fall on the method acting side of the roleplaying spectrum. This is totally a personal preference, and while it's my preferred mode of play, I have absolutely enjoyed games with people who are not comfortable with this. I'd like my fellow players to do this, but it's not essential to my fun, and the cooperative storytelling is far more important to me.
That's kind of the elevator pitch. You can stop here. But there's more philosophy, if you're interested.