Very basic description.
That round part under the Royal Enfield logo in that cutaway diagram is the clutch. It has part of it which is connected to that chain from the crankshaft which you can see in the diagram, and the other part is connected to the transmission shaft which goes inside and holds the gears. There is a powerful spring and friction plate system which clamps these parts together when the bike is riding, and the clutch lever on your handlebar releases this spring pressure when you pull it in, so you can change gears, and then lets it clamp again when you let out the clutch lever.
Inside the transmission, there are two shafts with various gears(cogs) on them which have different diameters and have gear teeth numbers to suit their size. When you shift gears, the shifter moves these gears back and forth on the shafts so that certain gears mesh together, to give you a certain gear ratio. Then when you switch gears again, different gears mesh together to give another gear ratio, and so on, all the way up to top gear.
The final drive chain is driven by the output shaft of the transmission, which is concentric with the main shaft, and is locked to the gear that is selected inside the gearbox, and that drives the output sprocket on the front of the final drive chain. The chain then drives the rear sprocket at the wheel, and these different size sprockets form the final drive gear ratio with the drive chain.