(note: For the historical formality, the final form of his equations were not written by him, but are an evolution of what he wrote in his works).The equations are kind of encrypted for anyone who is not familiar to vector mathematics. Even their meaning are kind of obscure to who doesn't know physics. But what they describe is the nature of magnetism, electric field and light. And more than this: it says that one is the result of the changing on the other.
So, take your little magnet you have in your refrigerator. Around it, we have a magnetic field in a certain form, shape and strength. Now, start shaking it! Yeah! Just like this: hold it in your hand and shake it as fast and hard as you can. What you are doing is changing the magnetic field in space, from left to right, right to left. What the equations of Maxwell says is that this change creates electromagnetic waves in the space, and those waves are... light.
But I cannot see a thing, you would say. And I would be very worried if you said you could see something. The reason you cannot see anything happening is that the strength and frequency of the changes you are provoking in the electromagnetic field are way to small for your eyes (or even an electronic device) to notice. Electromagnetic waves, as any other kind of wave, may come in low frequency (as your shaken magnetic) or very high frequency, or anything in between. Here goes a table that shows more or less the names we call every kind of electromagnetic wave:

The names and the division between the different frequencies are pretty much arbitrary and are defined more or less by the way those radiations (which are all from the same electromagnetic nature) interact with mater.
So, here we have: low frequencies (left in the graph) are called radio and TV waves. If you could shake your little magnet more then 1.000.000 (10e6) times a second, you would interfere with your radio reception. If you could shake it more then 10e14 (a 1 with 14 zeros after it) times a second, you would probably loose your hand, but you could then see visible light. Higher frequencies are ultra-violet, x-rays and gama-rays.
So, the "stuff" your radio catches from the air through the antenna, is, in nature, the very same "stuff" your lamp radiates. The difference is in the frequency of those waves. Our eyes can only perceive a little part of this whole spectrum (which we call visible light).
The most interesting part of Maxwell studies is the fact that he tried to make an understanding of the magnetic phenomena, and he ended up finding out the astonishing conclusion that magnetics and light are intricately connected.
This is profound. When two totally unexpected different fields of physics (or any knowledge field) are found to be connected, it brings a huge feeling of order in nature. Sometimes, it brings people to do stuff like this:

Here is a great link for a detailed explanation and meaning of the 4 equations. No need for previous knowledge, but it will sure helps a lot.

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