The way

What does this phrase mean: he has his way with words/colors,etc.?

5b -- Merriam-Webster.

5 a : characteristic, regular, or habitual manner or mode of being, behaving, or happening Knows nothing of the ways of women. b : ability to get along well or perform well. She has a way with kids. A way with words.

It's an ability, often natural rather than from training, to do something well.
A way with words; can express things well in writing or speech.
A way with kids; can establish a good relationship with children.
Not to be confused with 'got away with' meaning escaped without punishment.
He got away with murder.

a 'way' suggests a route through to success -to be in a mind-set - 'do' in Japanese, as opposed to merely having the technique 'jutsu'/'jitsu'.

ju-do / ju-jitsu (some spell it jutsu)
ken-do / ken-jitsu
bu-do, karate-do etc

L