http://en.wikipedia.org/wiki/Classical_electromagnetism_and_...

The main thing is to consider how charge densities change as we switch reference frames. Because the lengths contract when we switch to a reference frame where things are in motion, we see higher charge densities (same charge per less length).

A simple demonstration: Say we have a charge neutral, current carrying wire. We model it as a bunch of positive charges staying still, and some negative ones moving. The positive and neutral balance, but there is still a current because only the negatives are moving. Now imagine we switch to a different reference frame, one where the negative charges aren't moving, but the positive one's are, we are flying along parallel to the current. Now the positive charges contract relativisticly, so the density of positive charges is now greater than the negatives, and hence the wire appears to be carrying a net charge.