If you are not well-acquainted with special relativity, there is no way to truly explain this phenomenon. The best one could do is give you rules steeped in esoteric ideas like "electromagnetic field" and "Lorentz invariance." Of course, this is not what you're after, and rightly so, since physics should never be about accepting rules handed down from on high without justification.
The fact is, magnetism is nothing more than electrostatics combined with special relativity. Unfortunately, you won't find many books explaining this - either the authors mistakenly believe Maxwell's equations have no justification and must be accepted on faith, or they are too mired in their own esoteric notation to pause to consider what it is they are saying. The only book I know of that treats the topic correctly is Purcell's Electricity and Magnetism, which was recently re-released in a third edition. (The second edition works just fine if you can find a copy.)
A brief, heuristic outline of the idea is as follows. Suppose there is a line of positive charges moving along the zz-axis in the positive direction - a current. Consider a positive charge qq located at (x,y,z)=(1,0,0)(x,y,z)=(1,0,0), moving in the negative zz-direction. We can see that there will be some electrostatic force on qq due to all those charges.
But let's try something crazy - let's slip into qq's frame of reference. After all, the laws of physics had better hold for all points of view. Clearly the charges constituting the current will be moving faster in this frame. But that doesn't do much, since after all the Coulomb force clearly doesn't care about the velocity of the charges, only on their separation. But special relativity tells us something else. It says the current charges will appear closer together. If they were spaced apart by intervals ΔzΔz in the original frame, then in this new frame they will have a spacing Δz1−v2/c2−−−−−−−−√Δz1−v2/c2, where vv is qq's speed in the original frame. This is the famous length contraction predicted by special relativity.
If the current charges appear closer together, then clearly qq will feel a larger electrostatic force from the zz-axis as a whole. It will experience an additional force in the positive xx-direction, away from the axis, over and above what we would have predicted from just sitting in the lab frame. Basically, Coulomb's law is the only force law acting on a charge, but only the charge's rest frame is valid for using this law to determine what force the charge feels.
Rather than constantly transforming back and forth between frames, we invent the magnetic field as a mathematical device that accomplishes the same thing. If defined properly, it will entirely account for this anomalous force seemingly experienced by the charge when we are observing it not in its own rest frame. In the example I just went through, the right-hand rule tells you we should ascribe a magnetic field to the current circling around the zz-axis such that it is pointing in the positive yy-direction at the location of qq. The velocity of the charge is in the negative zz-direction, and so qv⃗ ×B⃗ qv→×B→ points in the positive xx-direction, just as we learned from changing reference frames.