I apologize for asking the duplicate question, but I still get confused after reading the related post. In particular, for the statement

if universe is represented by manifold $M$ with matter fields $\psi$ and metric $g$, and $\phi: M \rightarrow M$ is a diffeomorphism, then sets $(M, g, \psi)$ and $(M, \phi^*g, \phi^* \psi)$ represents the same physical situation.

what does "same physical situation" really mean? Does it have some mathematical formulation?

Also, in https://physics.stackexchange.com/a/596779, it said that we can actually prove the diffeomorphism invariance of the Hilbert action, can anyone gives a derivation?