For someone who only studied electromagnetism, what is the modern way to explain electromagnetic fields?

After reading most of the electromagnetism chapters of Feynman's lectures on physics, I would like to understand in more detail, at least an idea, of what causes the electromagnetic fields. Not sure where to go.

For instance, an electric field is caused by a charged particle. The presence of another particle would mean a force would act upon it. But is there a better explanation for this force to occur? By wikipedia browsing, photons are "the force carrier for the electromagnetic force". What does this mean exactly?

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The force acts between bodies, the electric field determines the force into the equations of motion of bodies. The body interaction is generally retarded so the field depends on time in a retarded way. The total electric field of a given charge is a sum of a "near field" that depends on time, but decays with distance as $1/R^2$ (they say it is "attached" to the charge), and a propagating field decaying as $1/R$. The latter corresponds to real photons (waves propagating with light velocity) whereas both are involved into the charge interaction. Depending on the value of $R$, one or another term may dominate.