Simplifying Complexity, One Answer at a Time!
Rules
A muse is someone who inspires them to do the thing. They don't necessarily have to do anything for the other person to be inspired by them.
An example could be an artist who is inspired to paint his favourite model numerous times in different lights and conditions as they just seem to be the right fit for them.
Or the songwriter telling songs about love or heartbreak from their own perspective and history as they had that with someone.
The muse of programming: https://en.wikipedia.org/wiki/Rubber_duck_debugging (sometimes just talking things out inspires)
Or as I have heard, getting expert advice.
Talking the problem through is sometimes the best way to understand what is wrong
https://en.wikipedia.org/wiki/Muse#Functions
But anyway, it's mythology, it's irrational, there is no need for an explanation. You're seeking a "scientific" or modern explanation of how it works, but it's not for you actually.