I’m a simracer, but I want to learn more about cars and eventually be able to work on them to some extent.
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A manual gearbox have two main shafts, an input and an output shaft. There are cogs on these shafts of different sizes which can slide on the shaft but rotate with it. The gear selector is mechanically moving these cogs so that on set of cogs mesh. This locks the ratio of the input and output shaft to a fixed value called the gear ratio.
When you change gear you first have to remove forces and mass from the system. So you disengage the clutch to disconnect the engine from the input shaft. Then you put the gear selector in neutral. The input shaft is now rotating completely on its own innertia and is very light weight. So when you select the next gear and the cogs of the new gear start to mesh a set of syncronizing rings on the cogs will mesh first to change the speed of the input shaft such that the cogs will mesh perfectly. Once you put the car in gear the cogs mesh fully, you can often feel it click into place from the initial syncronizing ring engagement to the cogs, the input and output shafts are in this locked ratio. Then you start engage the clutch to connect the engine to the input shaft again. The clutch is allowed to slip a bit when it is half engaged so that the speed of the engine and the input shaft on the gearbox is allowed to syncronize.
Specifically about race cars the syncronizing rings and the clutch are technically not needed to change gear in a car. But getting the gears to mesh without these is practically impossible for a novice driver and even hard for most experienced drivers. However using the sycronizing rings and clutch does wear them out and lose you power. For a race car you also have the issue that the engine can be too powerfull for the clutch and if abused it can ruin the clutch. For this reason racing drivers tend to learn how to match the engine speed and car speed with the gear ratio of the gear they switch to and make sure to have as little throttle application as possible when selecting the gear and using the clutch so the forces through the clutch and syncronizing rings are as low as possible. There are even real examples of race cars that have had clutch failures during a race and the driver still finished at a good pace and even some rare cases of clutches not being installed into some race cars to improve performance.
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