This idiom or expression like phrase has to do with:

Doing things in line, or associated, with someone or something.

When someone says “A famous soul singer plans to go country,” they mean “She wants to do country music.”
When someone says “Voters are going Republican,” they mean “Voters are starting to vote for Republican candidates.”
When someone says “Companies go green,” they mean something like “They are doing things in line with saving natural resources.”
Etc.