The tutorial explains how to implement asynchronous operations in an engine that initially runs synchronous filesystem I/O, which limits the speed of serving client requests. By using the server API available in the `create_instance` function, it introduces asynchronous methods to prevent blocking the engine API during `get` and `store` operations. The tutorial emphasizes that the design is focused on explaining the engine API rather than optimizing efficiency, suggesting potential improvements like using a thread pool instead of creating new threads for each task due to the overhead involved. It details the process of dispatching requests to new threads, using `notify_io_complete` to inform the memcached core when operations are complete, and handling errors if they occur. The code snippets provided illustrate how to update the `fs_get` method to run asynchronously and store results, with a structure for tasks that includes engine pointers, cookies, and operation-specific data.