Presumably the author is using tokio which requires the future constructed (e.g the async function) to be Send (either because of the rules of Rust or annotated as Send) since tokio is a work-stealing runtime and any thread might end up executing a given future (or even start executing and then during a pause move it for completion on another thread). std::sync::MutexGuard intentionally isn't annotated with Send because there are platforms that require the acquiring thread be the one to unlock the mutex.
One caveat though - using a normal std Mutex within an async environment is an antipattern and should not be done - you can cause all sorts of issues & I believe even deadlock your entire code. You should be using tokio sync primitives (e.g. tokio Mutex) which can yield to the reactor when it needs to block. Otherwise the thread that's running the future blocks forever waiting for that mutex and that reactor never does anything else which isn't how tokio is designed).
So the compiler is warning about 1 problem, but you also have to know to be careful to know not to call blocking functions in an async function.
One caveat though - using a normal std Mutex within an async environment is an antipattern and should not be done - you can cause all sorts of issues & I believe even deadlock your entire code. You should be using tokio sync primitives (e.g. tokio Mutex) which can yield to the reactor when it needs to block. Otherwise the thread that's running the future blocks forever waiting for that mutex and that reactor never does anything else which isn't how tokio is designed).
So the compiler is warning about 1 problem, but you also have to know to be careful to know not to call blocking functions in an async function.