First things first: Edison's light bulb had a filament, it was not an arc lamp. The Arc lamp was crucial to the invention of the arc transmitter. Actually early on they were more or less the same device. the key difference was that a true transmitter transmits on a specific frequency (more or less) and that ideally it be designed to convey data. early designs had trouble with both of these things. The 1888 book Municipal Lighting has a nice quote to start us off.
"For many years it has been known that an extremely brilliant light could be produced by slightly separating two pencils of carbon, through which a powerful current of electricity, was passing, as the mysterious force spans the gap with an “arc” of intense light. The one insuperable bar to the general introduction of this light, was its great cost, due to the necessity of producing the current by the consumption of zinc in the galvanic battery. In spite of this expense the arc light early found a limited application to lighthouses, and other important government works."
The British chemist Humphrey Davy is usually credited with inventing the arc lamp. But we don't know exactly when Davy made his discovery. Different sources cite anywhere from 1802 to as late as 1809. (The later dates are more credible) But in all versions he used charcoal strips as electrodes and connected them to a battery. The arc he then generated between those two terminals produced an intense light. But the effect had been known prior to him. It was probably encountered by several other electrical experimenters but it was documented at least as early as 1802 by Vasily Vladimirovich Petrov. He was a Russian scientist working on copper-zinc batteries he described his arc as a "special fluid with electrical properties." Davy's light was brighter, and his work is what made artificial lighting seem practical, and commercially viable.
Here's how it works. The arc is the discharge that occurs when a gas is ionized. In modern arc lamps that can be neon, mercury, argon, xenon, krypton, or sodium.then high voltage is applied and the tips of the carbon rods are heated to incandescence, generating light. Modern arc lamps use tungsten instead of carbon. In Davy's arc lamp, the electrodes are carbon rods in free air, his electrodes need to be in contact to arc. The electric current actually heats the terminals and maintains the arc across the teeny tiny gap. In modern lights it's different because we use a ballast to match the current etc.
The first arc lamp was not suitable for street lighting; it had a few problems. The electrical requirements were too high. Generator improvements reduced that problem but the electrical arc heating up the carbon electrode burned it away. Eventually the gap was too wide for the arc to jump. That was resolved by Paul Jablochkoff. In 1870, he devised a lamp that used two parallel carbon rods lengthening their life.
The only real remaining problem after that was this damn humming noise—a byproduct of the generated sparks. Several inventors seized on those implications: Nikola Tesla in 1886, Elihu Thomson in 1892, William Duddell in 1897, Reginald Fessenden as early as 1900... the history of radio rolls on from there. You can revisit this 19th century technology courtesy of a how-to video from MAKE: