A current in a wire creates its own magnetic field. When the wire is placed in the field of a magnet, the two fields interact, and this interaction is what produces the force on the wire. Magnetic forces are always due to the interaction between magnetic fields. The field of the wire interacts with the field of the magnet.
On one side of the wire the field of the wire points the same way as the field of the magnet, so the two fields add and the field is stronger. On the other side the fields point in opposite directions, so they partly cancel and the field is weaker. The wire is pushed from the stronger field towards the weaker field.
The same idea explains the forces between two magnets. Their fields interact, so like poles repel and unlike poles attract.
