If you place the wire near a permanent magnet, this temporary magnetic field interacts with the permanent magnet field. You will know that two magnets placed near one another either attract or repel. In the same way, the temporary magnetism around the wire attracts or repels the permanent magnetism from the magnet, and that is what causes the wire to jump.
Suppose you take a length of ordinary wire, make it into a big loop, and lay it between the poles of a powerful, permanent horseshoe magnet. Now if you connect the two ends of the wire to a battery, the wire will jump up briefly. It is amazing when you see this for the first time. It is just like magic! But there is a perfectly scientific explanation. When an electric current starts to creep along a wire, it creates a magnetic field all around it.