In physics there are chains of reasoning. These chains seek to uncover the causes of given effects, and to "unpack" the complex mechanism or structure which combines together to produce given results. Such a chain of reasoning has to reach to things and structures which are deemed "simpler", and then on to things "simpler still" but eventually it has to stop at something where we have no further explanation to offer. All we can say is "this set of ideas is logical and elegant, and fits the observed behaviour". In the case of electric and magnetic effects this basic set of ideas is called quantum electrodynamics, which itself is part of the structure of a theory called "the standard model" and there are hints that this model is itself part of a rather complex set of ideas called string theory.
To answer basic questions in physics, therefore, all we can do is point to aspects of these basic descriptions and say "well this is part of the basic structure; that is what the universe is like. The best we can do is describe it in detail and show how it all hangs together neatly."
Questions about how charges produce fields, and how fields produce forces on charges, are like this. They are asking about aspects of the behaviour which are close to the fundamental structure of ideas.
In the case of a magnetic interaction, a moving charged particle, or one carrying a property called magnetic dipole moment, gives rise to a magnetic field near to it, and this magnetic field cannot suddenly disappear, so the field near the particle joins smoothly to the field further away, and so on, until eventually the field encounters some other charged particle, and exerts forces on it. So the right way to think of this is not to focus only on the particles, but also allow that the fields themselves are part of the physical setup. It is a bit like the way a ship at sea can influence another ship by causing waves or a wake, or just making the water flow differently. In the ship analogy the water is not just a by-stander; it is part of the system. In a similar way, a magnetic field is part of the magnetic system.
But I have not yet said how a charged particle can give rise to a field. It is harder to do that. The best answer turns out to be "it just does", but we can say more about it by offering precise mathematical statements.
In the case of electric and magnetic effects the basic idea is that there is mutual interaction between electromagnetic fields and charged particles. We cannot answer the question "why is there this mutual interaction". It is just a basic property. It is the meaning of the term "electric charge". If the interaction happens then we say there is electric charge present. If there is no interaction then we say there is no electric charge. The theory then tells us further properties of this charge, especially the interesting ones that it cannot suddenly appear out of nowhere, nor disappear (charge conservation); it exists along with mass; the amount of charge does not change when you observe it from different inertial reference frames. Also, the total combination of electric and magnetic fields around a charge depends on how the charge is moving. What we can say is that this is not just a hotch-potch of ideas, like a collection of random articles found in the street. Rather, it all remains consistent with a remarkably small collection of mathematical equations, called the Maxwell equations and the Lorentz force equation (when quantum effects don't need to be considered in detail), or the equations and methods of quantum electrodynamics which underpin the Maxwell and Lorentz equations.
These equations can themselves be obtained by a general method called the Lagrangian method, applied to a fairly simple starting point, so this makes us judge that the overall framework can be called "simple" in the sense of "basic". (It is not simple in the sense of "easy to learn"; the mathematical methods are quite advanced.)
So, to conclude, to ask "why does a charge produce an electric and magnetic field?" is like asking "why is 2 the square root of 4?" The answer to the mathematical question is "because it just is", but we could add that $2 \times 2 = 4$. Similarly, a charge produces a field because that is the nature of what charge is; it is the name we give to a basic aspect of the physics. But to fill it out we can offer equations showing how the connection between charge and field looks in various cases, such as the equation
$$
{\bf E} = \frac{q}{4 \pi \epsilon_0 r^2} \hat{\bf r}
$$
for the electric field around a static point charge. The equation for the magnetic field gives a non-zero result either when the charge is in motion, or when the particle in question has magnetic dipole moment.
Ultimately the right way to think of magnetic and electric fields is not to regard them as separate, nor as if one causes the other, but rather they are different aspects of a single field called the electromagnetic field. This is bit like the way we can describe vectors in terms of one set of components or another, except here the field vectors are themselves components of a mathematical object called a tensor.
