To "dig" something is a slang term for thinking it is cool, or liking it. "I really dig this movie" = I think this movie is great. "I dig guys with cats" = I like guys who own cats. "I'm digging this look" means that you think the "look" (the appearance) is cool or attractive. The phrase could be used sarcastically or seriously, so it's not clear from your example whether the man actually likes how he looks without eyebrows, or is making a joke about how bad it looks.

Gray is correct. I would also add that the word is not used very much anymore where I live (west coast of the USA). It was quite popular in the 1970's in the USA.