What happens inside the space between two strong magnets?

You could actually try this, and see if it works for you. Be sure to include the hidden button battery and switch.

the system as shown will not work. The videos are fake.

An electric charge moving across a magnetic field is pushed in a direction that is at right angles to both the velocity and the field. In the apparatus pictured, as the coil is moved across the field, free electrons in the coil are pushed along the wire. When entering or leaving the field, the push on one side is greater than on the other. This produces a current which can do external work.

In the photo you provide the coil is partially inside the field and partially outside the field.
When you move the coil perpendicularly to the field the AREA exposed to the field either increases or decreases giving rise to an EMF.

Now if you ALSO permit the current to flow through a motor then you have a current across a magnetic field which creates a force.
That force opposes the motion so you must pull the coil with a force through a distance to move it.

The result is that the electrical energy created is coming from the energy that you give to the coil.

In other words a magnet/coil system CONVERTS mechanical energy into electrical energy. Just as in every generator/alternator on the planet.

It is most certainly NOT free energy, magnetic energy, or creating anything at all.

Electric field can be generated only when the magnetic field is moved with a velocity or magnetic flux changes in time. If these are absent, then no field is generated and bulbs do not glow. To run a dc motor, you need to rotate the coil in the plane which is here either x or y axis assuming the magnetic field to be along z axis. There in the above, the dc motor will not work and bulb will not glow.

What you are missing is that that they are tricks for entertainment or misinformation / trolling. YT is full of bollox like this.