I believed any kind of asynchronous execution makes a thread in invisible area.
This is your problem - this actually isn't true.
The thing is, your whole computer is actually massively asynchronous - requests to RAM, communication via a network card, accessing a HDD... those are all inherently asynchronous operations.
Modern OSes are actually built around asynchronous I/O. Even when you do a synchronous file request, for example (e.g. File.ReadAllText), the OS sends an asynchronous request. However, instead of giving control back to your code, it blocks while it waits for the response to the asynchronous request. And this is where proper asynchronous code comes in - instead of waiting for the response, you give the request a callback - a function to execute when the response comes back.
For the duration of the asynchronous request, there is no thread. The whole thing happens on a completely different level - say, the request is sent to the firmware on your NIC, and given a DMA address to fill the response. When the NIC finishes your request, it fills the memory, and signals an interrupt to the processor. The OS kernel handles the interrupt by signalling the owner application (usually an IOCP "channel") the request is done. This is still all done with no thread whatsoever - only for a short time right at the end, a thread is borrowed (in .NET this is from the IOCP thread pool) to execute the callback.
So, imagine a simple scenario. You need to send 100 simultaneous requests to a database engine. With multi-threading, you would spin up a new thread for each of those requests. That means a hundred threads, a hundread thread stacks, the cost of starting a new thread itself (starting a new thread is cheap - starting a hundred at the same time, not so much), quite a bit of resources. And those threads would just... block. Do nothing. When the response comes, the threads are awakened, one after another, and eventually disposed.
On the other hand, with asynchronous I/O, you can simply post all the requests from a single thread - and register a callback when each of those is finished. A hundred simultaneous requests will cost you just your original thread (which is free for other work as soon as the requests are posted), and a short time with threads from the thread pool when the requests are finished - in "worst" case scenario, about as many threads as you have CPU cores. Provided you don't use blocking code in the callback, of course :)
This doesn't necessarily mean that asynchronous code is automatically more efficient. If you only need a single request, and you can't do anything until you get a response, there's little point in making the request asynchronous. But most of the time, that's not your actual scenario - for example, you need to maintain a GUI in the meantime, or you need to make simultaneous requests, or your whole code is callback-based, rather than being written synchronously (a typical .NET Windows Forms application is mostly event-based).
The real benefit from asynchronous code comes from exactly that - simplified non-blocking UI code (no more "(Not Responding)" warnings from the window manager), and massively improved parallelism. If you have a web server that handles a thousand requests simultaneously, you don't want to waste 1 GiB of address space just for the completely unnecessary thread stacks (especially on a 32-bit system) - you only use threads when you have something to do.
So, in the end, asynchronous code makes UI and server code much simpler. In some cases, mostly with servers, it can also make it much more efficient. The efficiency improvements come precisely from the fact that there is no thread during the execution of the asynchronous request.
Your comment only applies to one specific kind of asynchronous code - multi-threaded parallelism. In that case, you really are wasting a thread while executing a request. However, that's not what people mean when saying "my library offers an asynchronous API" - after all, that's a 100% worthless API; you could have just called await Task.Run(TheirAPIMethod) and gotten the exact same thing.