The magnetic force on a current-carrying wire through a magnetic field is given by F = IL x B. When a current is passed through a magnetic field, the magnetic field exerts a force on the wire in a direction perpendicular to both the current and the magnetic field.
That is the idea here with the jumping wire. A wire is strung through a magnetic field. When the wire is hooked up to a battery, a current runs through the wire, and the part passing through the magnetic field is acted upon by an upwards or downwards (depending on orientation) force.
This is the set-up:
And here it is jumping: