The Round Earth Theory begins with the Greeks in 600BC, when they speculated that the Earth was round as a philosophical maxim to preserve the Grand Perfection of the Universe. Later, a fellow by the name of Eritosthenes in the first century BC.
Eristosthenes noticed that sunlight shining down water wells in Greece and Egypt at the same time on the same day had different incident angles. Using the angles he derived from these observations, he was able to determine that the Earth was a sphere of 250,000 stades, which happens to be about precisely 10% too large. Turns out he was fooled by the Egyptian government (naturally) which added 10% to its measurements of land for tax purposes.
Other Greeks, such as Claudius Ptolemy, not only determined the the Earth was round, but that it orbited the Sun. Yes, the heliocentric theory far and away predates Copernicus. The Greeks weren't Jews or Christians back then, so they didn't have a theological imperative to have the Universe revolve around the Earth.
The Round Earth Theory was further proved by Galileo Galilei and his infamous telescope. He watched several moons orbit an obviously round Jupiter, and that Mercury and Venus had phases like the Moon, meanings they were spheres orbiting inside the Earth's orbit, while the other planets did not have crescent phases, meaning they were outside the Earth's orbit.
Magellan, who was the first documented person to circumnavigate the globe in the early 1500s, put the final cap on the argument. Setting out from Europe heading west and returning from the east. Furthermore, there were a whole host of constellations (not to mention the Magellanic Clouds) that couldn't be seen from the northern hemisphere.
There was little doubt after that, though photos of Earth from space, obtained for the first time in the 20th century, proved beyond doubt that our planet was round. These photos view Earth from every position in the Solar System between the orbits of Mercury and Saturn.
Despite this long history of proofs, the Flat Earth Theory re-emerged in modern times in the 1800s, and has had several incarnations and reboots since then. These groups re-interpret, or more commonly, misinterpret scientific data to prove that the Earth is flat. The more extreme versions of the theory propose a vast conspiracy to make people think the world is round, though I have yet to find a viable reason why this would be important. While I am open-minded about such things, I prefer a bit more compelling evidence to take them seriously.
Like all things that deny reality, the Flat Earth Theory is rather easy to disprove, and the proof is rather pragmatic, as well.
Modern microwave communications quite easily disproves a flat Earth. Microwave technology is Line of Sight (LOS), meaning that the transmitter and receiver must be at the same levels or the signal will miss the target. On a flat surface, a microwave signal at wattages used in video transmissions - such as live news feeds - can travel quite a few miles and remain coherent.
On the curved surface of the Earth, the beam will eventually intersect the ground level if the receiver is more than 5 miles away. Microwave LOS transmitters therefore use tall towers spaced at regular intervals, called repeaters, to take the signal and resend it to the next tower. If you look closely at these towers, you will notice white vertical boxes mounted facing in various directions. Those are the transmitters. The round objects, usually with a conical cover on them, are the receivers. If you were to climb the tower with a pair of binoculars and stand behind any one of them (not recommended for health reasons), you would see about half of the next tower above the horizon with the paired transmitter or receiver facing directly at you.
In none of the videos I have watched on Flat Earth have I seen this very simple test mentioned, probably because it is rather hard to refute. LOS is LOS, and on a curved surface, the line will eventually intersect with the ground. Believe me, if communications companies could spend a lot less on those towers, they would. Grand conspiracies crumble in the face of economics and profits.
Having spent a significant part of my life aligning, bouncing and aiming microwave transmitters, I can attest that the world is round. I accept that some footage of Earth from the Moon missions may have been faked for a number of good reasons, but a flat Earth is definitely not one of them.
One has to wonder why such theories get promulgated so easily. I can envision a conspiracy to promote these kinds of things to create an air of lunacy around the whole area of inquiry. The point of this exercise, though, is to highlight the fact that often nonsensical theories like this are promoted in order to discredit the whole category of alternative inquiry. People will usually not persist in things that will get them ridiculed. However, discoveries are frequently made by people who step beyond the boundaries of accepted knowledge, and all it takes, really, is a willingness to change your perspective. Many of the household names today were crackpots in their own time. No one is laughing now.
By the way, if you are not sure that men landed on the Moon, then look no further than the dust kicked up by the Lunar Rovers. Yes, they really landed on the Moon. The big question is, who exactly were the men who went there?