How electromagnetic wave was related to visible light? The presentation I see in the textbooks goes something like this, Maxwell realized changing electric fields produce changing magnetic field and vice versa. Therefore, at least intuitively, one can understand why there should be such a thing as Electromagnetic wave. Which then Maxwell showed, as far as I know, by using the set of equations named after him that there indeed must be electromagnetic waves. And then the text books, at least the ones I have consulted, go on to declare that light is an electromagnetic wave without feeling much of a need to explain why that might be.
The identification of light with electromagnetic wave, I feel, is among many bits of physics that is taken for granted in the text books. I believe it is the result of a hindsight confirmation bias, that is we have so many consistent evidence that light is electromagnetic wave, the books don't think it necessary to explain. And for students, as they are hearing the statement from the elementary school, they often don't ask why must that be but accept it without questioning. I did so. And some other students I spoke to did that as well (forgive me, my sample size is not large enough regarding this, but I strongly believe this is the case for many students as well). It is not at all obvious to me that the existence of electromagnetic wave would imply that light is also such a wave. Maybe I am missing something crucial, but in this background let me ask the question in the following.
How, historically speaking, light was identified as an electromagnetic wave? What was the Logic/experiment that led scientists to suspect that light is an electromagnetic wave? (The logic can also be a vision, like in the case of Faraday who realized electric charges produce electric fields.) After searching in the internet I found this quote by Maxwell from "A Dynamical Theory of the Electro-Magnetic Field,"

We have strong reason to conclude that light itself—including radiant
heat and other radiation, if any—is an electromagnetic disturbance in
the form of waves propagated through the electro-magnetic field
according to electro-magnetic laws.

I basically want to know what is that strong reason that Maxwell is referring to and which has led him and other scientists to realize that light is a part of the electromagnetic spectrum. kindly weigh in.
 A: It was based on some experimental as well as theoretical measurements. Maxwell calculated the speed of his so called em waves for vacuum at that time using the formula derived from his equations
$$ V^2 = \frac{1}{\mu \epsilon}$$
The value which he got from the above equation was $3 × 10^8 /s$ and this value was very close to the experimental measurement of the speed of light which was $299,792,458 m/s$ .
I read somewhere that light is an em wave on this basis but may be there were other reasons.
This is the thing you are looking for I guess.
Hope it helps ☺️.
A: 
In 1856 with Rudolf Kohlrausch (1809–1858) he (Wilhelm Eduard Weber) demonstrated that the ratio of electrostatic to electromagnetic units produced a number that matched the value of the then known speed of light. This finding led to Maxwell's conjecture that light is an electromagnetic wave. This also led to Weber's development of his theory of electrodynamics. Also, the first usage of the letter "c" to denote the speed of light was in an 1856 paper by Kohlrausch and Weber.

About Wilhelm Eduard Weber on Wikipedia
Maxwell itself reported about the dependency between the electric permittivity and magnetic permeability:

From this I determine the mechanical force acting, 1st, on a moveable conductor carrying an electric current; 2nd, on a magnetic pole; 3rdly, on a electrified body.
The last result, namely, the mechanical force acting on an electrified body, gives rise to an independent method of electrical measurements founded on its electrostatic effects. The relation between the units employed in the two methods is shown to depend on what I have called the „electric elasticity“ of the medium, and to be a velocity, which has been experimentally determined by M.M. Weber and Kohlrausch.


This velocity is nearly that of light, that it seems we have a strong reason to conclude that light itself ... is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.

A Dynamical Theory of the Electromagnetci Field. By J. Cleark Maxwell
So your question was very reasonable.
A: The speed of light was measured before Maxwell with the Fizeau-Foucault apparatus:

*

*1848-1849 by Fizeau: $315000$ km/s

*1862 by Foucault: $298000$ km/s

*1872-1876 by Cornu: $300400$ km/s

Around the same timer (1861-1862) Maxwell set up his equations
for the electric and magnetic field.
By solving these equations he could predict the existence of electromagnetic waves.
And he could derive the speed of these waves from the constants
$\epsilon_0$ and $\mu_0$ appearing in these equations.
$$c=\frac{1}{\sqrt{\epsilon_0\mu_0}} \tag{1}$$
These constants had already been measured before from electrostatic
and magnetostatic experiments. See Vacuum permittivity and
Vacuum permeability.
$$\epsilon_0=8.85\cdot 10^{-12}\text{ As/Vm}$$
$$\mu_0=1.26\cdot 10^{-6}\text{ Vs/Am}$$
Maxwell plugged these numbers into (1) and found
$$c=\frac{1}{\sqrt{8.85\cdot 10^{-12}\text{ As/Vm}\cdot 1.26\cdot 10^{-6}\text{ Vs/Am}}}
=2.99\cdot 10^8\text{ m/s}$$
Surprisingly this was the same value as the speed of light measured with the Fizeau-Foucault apparatus.
Until then nobody could generate and detect
electromagnetic waves. Therefore nobody could
experimentally prove a connection between light and electromagnetic waves.
And now Maxwell found, these two things have exactly the same speed.
Therefore he conjectured that light is an electromagnetic wave.
Of course he did not know the frequency
of these electromagnetic light waves to prove his conjecture.
And hence it was still a speculation at his time.
Even later (1886-1887) Hertz was the first one to
generate and detect electromagnetic waves.
He used a spark transmitter and spark receiver, and measured
its frequency to be around $\nu=50\text{ MHz}$
and its wavelength to be $\lambda=6\text{ m}$.
From that he could determine speed of the wave to be
$$c=\lambda\nu=6\text{ m}\cdot 50\text{ MHz}=3\cdot 10^8\text{ m/s}$$
Here again he got the same speed as predicted by Maxwell
and as measured from light.
So the electromagnetic waves, which Maxwell predicted
theoretically, were finally confirmed experimentally.
