How did Maxwell discover that light is an electromagnetic wave?

Maxwell found that light is an electromagnetic wave by following his equations to an unexpected result. Around 1862 he calculated that an electromagnetic field propagates at approximately the speed of light. He considered that more than a coincidence, and in 1865 he showed that electric and magnetic fields travel through space as waves at the speed of light, and proposed that light is such a wave.

The key was a repair. Ampère's law said that magnetic fields relate to electric current. Maxwell's addition said that magnetic fields also relate to changing electric fields, which he called displacement current. Together with Faraday's finding that a changing magnetic field makes an electric field, this makes a loop: a changing magnetic field makes a changing electric field, which makes a changing magnetic field, and the perpetual cycle lets the wave move through space by itself.

The striking part is where that speed came from: it could be worked out from simple electrical experiments, with no light involved. Using the data of the time, Maxwell obtained 310,740,000 metres per second, close to the speed of light. In 1865 he wrote that light is "an electromagnetic disturbance propagated through the field according to electromagnetic laws". Light is one form of electromagnetic radiation, as are X-rays and radio waves.

He was building on Faraday's work. Maxwell introduced the concept of the electromagnetic field in comparison to the force lines that Faraday described. His waves were later confirmed: Heinrich Hertz first conclusively proved that the electromagnetic waves exist, the waves we now use for radio.