How could 19th century people know light is electromagnetic waves? From Maxwell's equation, we can find out that certain waves exist. However, it's unclear to me why 19th-century people thought that what they had called light is a wave.
As far as I know, 19th-century people weren't able to make light visible with electronic or magnetic devices of that era. So I wonder how they could connect these two together.
How could they know? Or why did they think so?
 A: As you already said, using all four Maxwell equations in a vacuum ($\rho=0$, $\mathbf{j}=\mathbf{0}$), we get the wave equations:
$$\Delta\mathbf{E}
=\nabla(\underbrace{\nabla\cdot\mathbf{E}}_{=0})
-\nabla\times\left(\nabla\times\mathbf{E}\right)
=\nabla\times\frac{\partial\mathbf{B}}{\partial t}
=\mu_0\varepsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2}$$
$$\Delta\mathbf{B}
=\nabla(\underbrace{\nabla\cdot\mathbf{B}}_{=0})
-\nabla\times\left(\nabla\times\mathbf{B}\right)
=-\mu_0\varepsilon_0\nabla\times\frac{\partial\mathbf{E}}{\partial t}
=\mu_0\varepsilon_0\frac{\partial^2\mathbf{B}}{\partial t^2},$$
which describe a wave propagating with the velocity $1/\sqrt{\mu_0\varepsilon_0}$, whose value is exactly that of the speed of light, hence $\mu_0\varepsilon_0c^2=1$. James Maxwell commented this result with: "This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electro­magnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws."
A: Not everyone did think light was a wave. Early in the 19th century, it was heavily debated. One experiment that convinced most people was the double slit experiment. Shining light through two narrow slits showed interference in the light. This is a property of waves, not particles.

This article goes into some more details on what inspired Thomas Young to do the experiment and provides some of the arguments for light being a particle. Reminder, light is a wave, but it is also a particle.
If you are specifically wondering how they proved light is an electromagnetic wave, That credit goes to Michael Faraday.

(He) discovered that the plane of polarization of linearly polarized light is rotated when the light rays travel along the magnetic field direction in the presence of a transparent dielectric, an effect now known as Faraday rotation.

That is what inspired Maxwell to do his experiments and studies on electromagnetic light.
A: Actually, the debate whether light is a wave or a flow of particles dates back to Descartes (wave theory, 1637) and Gassendi (corpuscular theory, around the same time). After that the wave theory was being developed, which culminated in Arago confirming in 1818 counterintuitive predictions of Fresnel's wave theory. This led to consensus in favor of the wave theory.
At this point people didn't actually know the nature of light waves and the medium they propagated in, although it was already known that the waves are transverse.
And in 1861-1862, a decade after Fizeau's measurements of the speed of light, Maxwell published his theory which connected light to electricity and magnetism.
A: To restate your question, one may ask, how do we know that light has electric and magnetic field vectors oscillating perpendicular to each other and perpendicular to direction of propagation, or, light as a transverse wave has two state of polarisation. One of the experiment that in a way concludes that light has two state of polarisation is the experiment of blackbody radiation. The entire spectrum energy density equation won’t match with experimental data if a scale factor of 2 is not considered. The reasoning behind this scale factor of 2 is that it comes from two state of polarisation of light as a wave.
This can be an indirect evidence that light itself is electromagnetic in nature, with electric and magnetic field vectors forming two states of polarisation of light waves.
