There is a deep reason that I don't fully understand having to do with special relativity. I'll give it a shot.
Maybe you have heard of something called length contraction. Essentially when an object moves very fast its length actually gets shorter. This was one of the consequences of Einstein's special relativity. Now think about a current flowing through a metal sheet. There is an equal amount of negative and positive charges (since atoms are neutral). The negative charges are moving, say, to the right. If you fasten yourself to one of those negative charges -- that is, consider the system from the reference frame of a negative charge -- it will look like the whole sheet is moving in the opposite direction while you are standing still. OK, the sheet is moving, that means, according to special relativity, that its length contracts. But the sheet has a net positive charge, so if its length contracts, then the charge density of the sheet must increase, and if the charge density of the sheet increases, then there is a higher concentration of positive charge per unit area than negative charge per unit area. Thus applying special relativity to this sheet of current implies that the sheet has a net positive charge, and what do net positive charges do? They repel other positive charges, of course. Remember, though, this was only from the perspective of the charges moving in the sheet. The upshot is that when you move along in the direction of the current, you observe a length contraction, and therefore an electric field perpendicular to the direction of the current. Prior to special relativity (and still today for practical purposes) this effect was called "magnetism."
Be patient, though, you'll get to this stuff.