The shape of the magnetic forces in and in the vicinity of the poles of a, let’s say, cylindrical magnet is twisted, very similar to a stranded wire.

(image borrowed from adko.eu)

When we lower the magnet on one side of a metal wire, then we can imagine something similar to two helical gears at angle of 90 degrees. The twisted field of the permanent magnet is the one gear; the induced magnetic field in the wire, which is also around the wire and is also spiral-shaped, is the other gear.
The twisted field of the permanent magnet is static when the magnet is not moving. When the magnet is moving, then its field is moving together with the magnet. This moving field we can call a magnetic wind. When we lower the magnet on one side of a metal wire, then this magnetic wind induces another magnetic wind in the wire, whose effect spreads also around it. This opposes to the first, just as a mechanical gear would offer a resistance to another gear by which is moved.
When we induce a current in a solenoid by moving a magnet in and out of it, then we have a device very similar to the mechanical “push and spin” devices, which can be found in kids toys, ashtrays etc.
Please see also this answer of mine on Quora
Is it possible to increase the current of a power source via an induced current?