As stated the question is a tautology. Charged atomic particles are charged because you just said they were charged.

"Atomic particles" is a little vague. I assume you mean "subatomic particles". Some of these are charged, and others aren't. For the fundamental particles, there's no known reason they have a charge (or not), they just do.

In addition to the electromagnetic charge (which is probably what you meant when you said "charged"), physicists need to consider a different kind of charge, which is variously called the strong nuclear or color charge. I think "color charge" is a fun phrase, so that's what I'll call it.

Electrons have an electromagnetic charge, but no color charge.

Photons have no charge.

Quarks have both a color charge and electromagnetic charge.

Gluons have a color charge, but no electromagnetic charge.

Those are fundamental particles, so the reason they have or don't have those charges is "because that's the way things are."

Protons are made of quarks (with electromagnetic and color charges) and gluons (with color charges). The color charges cancel out, so protons have no overall color charge, but the electromagnetic ones don't, so they wind up with an electromagnetic charge.

Neutrons are also made of quarks and gluons. However, in neutrons, both the color and electromagnetic charges cancel out, so neutrons have no overall electromagnetic charge and no overall color charge.

(Technically, there's also "gravitational charge," which is a measure of how much the particles interact with a gravitational field. However, everything, even photons, has a gravitational charge, so that's kind of boring.)