Magnetic forces are caused by the interaction between magnetic fields. Two magnets attract or repel because the field of one interacts with the field of the other.
A current-carrying wire makes its own circular magnetic field. When the wire is placed in the field of a magnet, the two fields overlap. Where the field lines of the two fields point in the same direction they add together and the field is stronger. Where they point in opposite directions they partly cancel and the field is weaker.
This gives a combined field that is strong on one side of the wire and weak on the other. The wire is pushed from the region of stronger field towards the region of weaker field, which is why it experiences a force. If the current is reversed, the wire's own field reverses, the strong and weak sides swap, and the force is reversed.
