The base mechanics that every RPG needs is some form of task resolution. Basically, you say "I want to do this" and you need to have a fair and impartial way of determining whether or not you succeed.

Imagine two children playing a make believe game of War. One says "Bang! I shot you, you're dead" and the other one says "No you didn't, you missed". What they need is a set of rules that determines what happened, because they can't decide. You could just toss a coin - heads, you succeed, tails, you fail. That's the root of RPG mechanics. It could be anything - convincing an evil prince to release some prisoners, or running down a bank robber, or trying to fly the Millennium Falcon out of the spaceport while under fire. You're just trying to see if you succeed or fail, and sometimes by how much.

Then it doesn't seem fair to have everything be binary, or that everyone is equally good at all things, so you split actions up into physical and mental. Then you say that this guy is really strong, but he's also good at poker. So now he has a skill. So maybe you take the best of five coin flips. Or you introduce something like dice that gives a wider range of outcomes.

And that's where things like stats and skills and all those mechanics come from. They're just granularity for coin flips for task resolution between two or more people playing make believe. They can be as simple or complex as you like.