What is the purpose of a transit visa, under what circumstances is is it required, and what happens when an individual arrives at an airport without one?
The purpose of a transit visa is… to transit through a country on the way to another destination. In some cases, a transit visa is simply a regular visa with a different name based on the purpose of the journey. When there is a distinction between a transit visa and a regular visitor's visa, there might be several differences:
In theory, I guess that someone landing without a transit visa where required could be refused entry and forced to return to their point of origin, be detained until they get their connecting flight or, if none of this is possible for one reason or another, face deportation. However, the rules are designed to make sure that it doesn't happen in the first place. Airlines are bound by international agreements to take back people who are refused entry and might face additional fines when it happens, depending on local law. This provides them with strong incentives to check if passengers fulfill any relevant visa obligation before boarding.
Note that in many cases, someone making it to a state's territory has the right to apply for asylum, even if they got there illegally or don't have a visa to enter the country. They could be detained and ultimately deported if their application is unsuccessful but (in principle) not immediately returned to their point of origin. Requiring transit visas for citizens of high-risk countries is one way potential destination countries make sure asylum seekers don't even get that far and limit the number of requests.