The purpose of creating Async methods is so you can Await them later. Kind of like "I'm going to put this water on to boil, finish prepping the rest of my soup ingredients, and then come back to the pot and wait for the water to finish boiling so I can make dinner." You start the water boiling, which it does asynchronously while you do other things, but eventually you have to stop and wait for it. If what you want is to "fire-and-forget" then Async and Await are not necessary.
Simplest way to do a fire and forget method in C#?
Starting a new task queues that task for execution on a threadpool thread. Threads execute in the context of the process (eg. the executable that runs your application). If this is a web application running under IIS, then that thread is created in the context of the IIS worker process. That thread executes separately from the main execution thread, so it goes off and does its thing regardless of what your main execution thread is doing, and at the same time, your main execution thread moves on with its own work.