I know that a charged particle is defined as a particle that experiences a force in an electric field, as explained in this post: What does a subatomic charge actually mean?
However, I still don't understand: Where does this force come from? Quantum mechanically, what does it mean for particles to repel or attract one another? Or more broadly, What does charge in general actually mean?
Charge means that the body experiences a force in an electric field. A charge generates an electric field, which generates a force on other charges particles. Two bodies are said to repel if they force each other away and two bodies are said to attract if they force each other closer together.
Now, I'm not really answering your question here of "why," I admit that. The problem is, it's not really possible to say why something like charge works the way it does. We can describe it using mathematical structures that represent (and even predict) the behavior of fundamental behavior of quantum objects, but when you start asking "why," well... They're called 'fundamental' because that's just the way it is. There's no reason like electric charges repel and opposite charges attract that we can even find a hint for. But they do, and we can use that.
According to Richard Feynman, the charge is the probability of a particle interacting by the electro magnetic force. More specifically it describes the amplitude of the "probability arrow" of a certain electromagnetic interaction taking place.
Much like @Asher has mentioned already, the standard model cannot provide an explanation for why certain particles interact via the electro magnetic force much like the standard model cannot explain why the particles that we know exist.
