What does mutual induction of electric and magnetic fields in an EM wave mean?

Before we begin to address the specifics of your question, we need to talk about fields. Fields are, loosely speaking, a number (vector) assigned to each point in space. In the realm of electrodynamics, we study two such vector fields, electric and magnetic. How do we decide on what vectors should each point in space be assigned? Well that’s what Maxwell’s equations tell us. We call each such possible assignment of values as a configuration. Rest of electrodynamics is to set up and study interesting configurations.

Right away you’ll see that any equation that “links” magnetic fields and electric fields (the curl equations) are not linking so locally, but over the entirety of space! Now you see that not all field configurations support endless propagation. It’s only certain special ones that do.

One such special configuration is one where the strength of the fields (norm of the vector) oscillates sinusoidally. These are called plane waves. And it turns out, every possible configuration can be expressed as a linear combination (sum) of different plane waves!

---

In summary, fields are values all throughout space. Maxwell’s equations restricts the possible values to be taken. Any relation between electric and magnetic fields are necessarily so all throughout space. And there are special configurations, like plane waves that support propagation.

I'll try to explain with little to no math.

Explanation of Relevant Maxwell's Equations:

According to Maxwell's Equations, an increasing electric field flux is associated with a magnetic field that curls around it (the direction of the curl is the direction the fingers of your right-hand curl when you point your thumb in the direction the flux is increasing in).

More precisely, the rate of change of the electric field equals the curl of the magnetic field (you can think of the curl as how much the magnetic field vectors rotate around each point in space).

$vec{nabla} times vec{B} = mu_0 epsilon_0 frac{partial vec{E}}{partial t} + mu_0 vec{J}$ (we're considering a case where $vec{J} = 0$)

Similarly, Maxwell's Equations say that an increasing magnetic field flux is associated with an electric field that curls around it. But this time the direction of the curl is the direction the fingers of your right-hand curl when your thumb points in the direction opposite to the direction of the increase.

$vec{nabla} times vec{E} = - frac{partial vec{B}}{partial t}$

How This Leads to Electromagnetic Waves

Now, if you start with an oscillating electron (say an electron in a radio antenna that's been forced to oscillate up and down) then that will produce an electric field that changes the direction it is pointing, oscillating up and down.

The oscillation will cause the flux of the electric field to switch the direction it's increasing in so then you'll have a magnetic field that is curling around the oscillating electric field. But the direction that magnetic field is curling in is changing (since the direction the electric flux is increasing in is changing).

But now, since the direction the magnetic field is curling is changing there is now an oscillating magnetic flux, this time through a plane perpendicular to the plane the electric flux was oscillating in.

The direction that magnetic flux is increasing in is also changing. That changing direction of the magnetic flux increase is associated with an electric field that curls around the changing magnetic flux and the direction of that curl is changing.

So now you have an oscillating electric field again and the process repeats.

To see more clearly what I mean in the How This Leads To Electromagnetic Waves section, watch this video (https://www.youtube.com/watch?v=SS4tcajTsW8) from 23:40 on. There's a nice animation of this concept in it.

If you want to see how the oscillating electron creates a changing electric flux/field in the first place watch this video (https://www.youtube.com/watch?v=DOBNo654pwQ).