There is a parallel in the history of physics that I think is helpful in illustrating wat Maxwell recognized.
In the Principia Newton posited that it is possible to derive the speed of sound from first principles.
Sound is oscillating compression and rarefaction of air mass. Oscillation requires presence of the following two properties:
There is a state of lowest strain, we can call that the equilibrium state, and when away from the equilibrium state there is a restoring force, towards the equilibrium state.
Something that has been put into motion will keep going, unless counteracted by a force. (In the case of mass referred to as 'inertia')
In the case of the speed of sound:
Newton argued: given the value of the density of air, and the value of the elasticity of air, both of which we can measure, we can derive the speed of sound.
For an example: a youtube video created by the University of New South Wales (Australia) physics department a presenting a derivation of the speed of sound
The bigger the restoring force, the faster the propagation speed.
The smaller the inertia, the faster the propagation speed.
In the years leading up to formulating the equations we refer to as 'Maxwell's equations' Maxwell had used a mechanical model of the Luminiferous Ether. Maxwell didn't assume that mechanical model was true, but he did use it as a guide for ideas on how to proceed. I won't go into the details of that mechanical model here.
One of the properties of that mechanical model is that it supported the concept that Maxwell called 'displacement current'.
If you have a capacitor in the form of two separated plates the substance that is in between the plates is called the dielectric. The composition of the dielectric has an effect on the physical properties of the capacitor as a whole. In particular, if the dielectric has ions that have some leeway to move relative to each other then the capacitor can store more electric energy.
If all the negative charge in the dielectric moves a bit to the right and all the positive ions move a bit to the left the overall effect is the same as that of overall current flow.
Maxwell pointed out that the known properties of capacitors indicate that even in the absence of any physical dielectric, that is, even when the space between the capacitor plates was depleted to a vacuum, measurement of the physical properties was consistent with a displacement current in the vacuum. (For more information see the wikipedia article about permittivity
Maxwell noticed that the properties that he had to craft into his mechanical model (of the Luminiferous Ether) in order to account for all known phenomena gave rise to the necessary conditions for oscillation.
When you strain a dielectric away from equilibrum there is a force that forces back towards equilibrium.
Maxwell's model also had a counterpart of inertia.
In the case of Maxwell's mechanical model of the Luminiferous Ether the restoring force is very, very large, leading to a very, very large speed of propagation. See also the wikipedia article about the Electromagnetic wave equation
Maxwell's mechanical model of the Luminiferous Ether cannot be physical reality, but clearly Maxwell was able to use it to great advantage
Wikisource has a transcript of the 1861 paper On physical lines of force
Proposition XVI states:
To find the rate of propagation of transverse vibrations through the
elastic medium of which the cells are composed, on the supposition
that its elasticity is due entirely to forces acting between pairs of
Maxwell finds a velocity of propagation of the transversal vibrations in the medium that is close to the known speed of light (314858 km/s, as measured by Fizeau). The 'Treatise on Electricity and Magnetism' was published in 1873, but the recognition of transversal vibrations propagating at the speed of light was already in the 1861 paper.