Let me try a slightly different approach.
How do we know light is an electromagnetic wave? Well, first, is there any reason to think it's a wave? Yup. See descriptions of interference patterns and polarization. These phenomena behave exactly as if light is a wave. So, why is it electromagnetic? This connection actually came about from the other end. James Clerk Maxwell produced his
famous equations and realized that, if you plug in some experimentally-determined constants (relative permeability and magnetic susceptibility), his equations can produce a propagating wave with a velocity exactly equal to that of light. This, of course, suggested that light is an electromagnetic wave - unless you are a big believer in coincidence.
For about 40 years, Maxwell's equations ruled. Then, those pesky experimental types started noticing that some aspects of light just don't behave as a wave should. The photoelectric effect is a good example. Various folk realized that something was wrong, and in 1905 a young patent clerk named Einstein (does the name ring a bell?) won the Nobel Prize for his paper which concluded that, at a small enough scale, light can also be thought of as particles. You'll note that his Special Relativity paper, also published in 1905, did not get the Nobel.
So, is light a wave or a particle? The answer is either a resounding yes or a thunderous no. It all depends on how you measure light. If you have a lot of energy being transmitted, waves are the way to describe it. If you're talking about really small amounts, quantization effects start to show up and it's better to think in terms of particles. And, just to mess with your mind, there are situations where, in effect, both are true. The classic experiment that deals with this is the double slit experiment and at very low light levels, you can detect single photons - but they appear at different places as if they were also waves. If this all seems pretty weird, don't worry. The physics community as a whole struggled for nearly 50 years with wave particle duality, and the problem is made worse by the fact that, at some level, it appears to apply to pretty much everything. Certainly you can get interference patterns from electrons, and they're particles, right?
But back to photons. The process you mention, emission by electrons changing energy levels, occurs at very, very small scales, and when looking at this level the particle description is the one that is most obvious and useful. There are various other ways to produce light (or at least EM radiation) which do not involve electrons changing energy levels. These include, for instance, electron/positron annihilation and nuclear fission and fusion. But producing light (or heat - infrared radiation) is common in our experience, so that's a useful source to consider. If the electrons are just bouncing around, you get essentially random wavelengths and this is seen in blackbody radiation. But if you can somehow constrain the way electrons can gain or lose energy, you can get light at very specific wavelengths, such as lasers. Explaining the specificity of these wavelengths is basically impossible with classic wave theory, but simple enough under quantum mechanics, which is the ruling model at atomic levels. Then, of course, if you've produced a whole lot of photons you get a beam of light which may well be usefully described (for lens design, for instance) as an electromagnetic wave.
The thing to keep in mind is an old saying (well, not that old) in science: All models are wrong, but some are useful