What does the word " muse" mean when a man calls a woman his "muse"?

His inspiration.

she inspires him

It means that he is inspired by her, usually to create something artistic. Check here and here.

My 2 second google search yielded this:
–noun
1. Classical Mythology .
a. any of a number of sister goddesses, originally given as Aoede (song), Melete (meditation), and Mneme (memory), but latterly and more commonly as the nine daughters of zeus and Mnemosyne who presided over various arts: Calliope (epic poetry), Clio (history), Erato (lyric poetry), Euterpe (music), Melpomene (tragedy), Polyhymnia (religious music), Terpsichore (dance), Thalia (comedy), and Urania (astronomy); identified by the Romans with the Camenae.
b. any goddess presiding over a particular art.
2. ( sometimes lowercase ) the goddess or the power regarded as inspiring a poet, artist, thinker, or the like.
3. ( lowercase ) the genius or powers characteristic of a poet

Yes, a source of inspiration, however, muses do not have to be people.

My muse is my pet goose.

My muse the silly gooz.

It means he is getting creative in his use of flattery to get into a woman’s pants.
His actual muse is “Horniness” XD

Muse are spirits/goddesses that give inspiration to people. They were first observed and documented in ancient Greece.
And yes, they are real.

I always thought it meant his “plaything”, something he uses to amuse himself.

I believe it to be a very romantic way to let a woman know that she is his inspiration and that she touches his soul.

Being in love always made me so much more creative and productive. Love is a creative power. The muses inspire creativity; so one’s beloved is often a muse.

The end result of being a muse to someone is that they are inspired to the point that it triggers their passion and creativity. But the reason a woman is considered a muse is because she encompasses all the qualities the artist considers perfect in a physical, spiritual or emotional way. The muse is the model of perfection in his eyes.
The muse-artist relationshp is unique. The muse brings out intense emotions on the artist and has a power over the artist to the point that sometimes he is lost without his muse. If the muse leaves, someone else —albeit having similar characteristics— just won’t do.

Outside art, if someone calls you their muse it’s beyond a term of endearment and it involves the muse being a positive factor in the thoughts, feelings and actions of the person who thinks you are his muse.

In general, it means that the woman is of great inspiration to him, whether it be artistically, emotionally or intellectually. Usually it refers to the arts, like painting or sculpting or poetry, or music composition.

But I have known of some cases where a fellow uses that term with a woman, because he knows she will be extremely flattered by it, and will do almost anything for him if he says that to her (if she already likes him). Hopefully he’s not just saying it to get the woman to sleep with him, because that would be mean.

In that particular case, it’s almost as bad as a man telling a woman he loves her (when he doesn’t) just to get her to sleep with him. Being called someone’s muse makes it seem like the person really really likes you. The only reason I am urging caution, is because one of my friends, who had an abusive boyfriend used this term and it really lured her in.

She brings out his creativity.

A friend of mine is a poet. When her husband died she talked about all the poetry she had written about him. The she talked about all the poetry she was still writing about him. I asked her if he was her muse. She cocked her head to the side and thought a moment, then said, “Yes, I guess he is!”

It works both ways, gender-wise.