So let's start with Purcell's basic explanation of magnetic forces. Shown below is a model of a wire with a current flowing to the right. To avoid minus signs I'm taking the current to consist of a flow of positive charges, separated by an average distance of l. The wire has to be electrically neutral in the lab frame, so there must be a bunch of negative charges, at rest, separated by the same average distance. Therefore there's no electrostatic force on a test charge Q outside the wire. What happens, though, if the test charge is moving to the right? For simplicity, let's say its velocity is the same as that of the moving charges in the wire. Now consider how all this looks in the reference frame of the test charge, where it's at rest. Here it's the negative charges in the wire that are moving to the left. Because they're moving, the average distance between them is length-contracted by the Lorentz factor. Meanwhile the positive charges are now at rest, so the average distance between them is un-length-contracted by the Lorentz factor. Both of these effects give the wire a net negative charge, so it exerts an attractive electrostatic force on the test charge. Back in the lab frame, we call this force a magnetic force.

By the way, it's remarkable that we can measure magnetic forces at all, since the average drift velocity in a household wire is only a snail's pace: v/c is typically only 10^-13, so the Lorentz factor differs from 1 only by about one part in 10²⁶. We can still measure this effect because the total charge of all the conduction electrons in a meter-long wire is tens of thousands of coulombs; two such charges separated by only a few millimeters would exert enormous electrostatic forces on each other.

You can repeat the preceding calculation for the more complicated case where the test charge's velocity differs from the average drift velocity of the charge carriers in the wire. Their velocities can also be opposite in direction. In each case, you still get the correct expression for the magnetic force on the test charge. The case where the test charge moves perpendicular to the wire is still more complicated, but we can understand it qualitatively after digressing to consider how electrostatic fields transform from one reference frame to another.

Information provided by: http://physics.weber.edu