Curious as to why this substitution of the root for a ♭VI chord works. It's pretty standard but I don't get the theory of it. I recently saw it being used in the context of a  ii - V - I to delay the final resolve of a song. Specifically the following:

|| A#11(13) Amin9 | C/D D7b9 | EbMaj7♯11(13) ||

I've seen other progressions walk up to resolve as such: ii - V - ♭VI - ♭VII - I

What is the theory surrounding a ♭VI substitution like this?