He is a kendo master.
She is one of the four haiku masters in this country.
Not only is he an expert at chess, he’s also a master at cards.
A person whose name is known (because he's famous?). Why is this person's name known? Why is this person famous? It's because they are an expert / master at something. How else does someone become famous or well-known? That's pretty much it, even if they're just an expert at being an idiot.
This is a jukugo word, which usually means on'yomi readings from the kanji. If you know the readings of your kanji you'll know how to read this as well.