If a politician is on-message, they keep saying what their party wants said. They stay on the same few ideas and the same phrases, especially when a journalist tries to pull them somewhere else. The phrase is a little unkind: it often hints that the person is not really thinking, just repeating the line. The opposite is off-message — they said something the party did not want said, and now there is trouble. You hear this in news about elections and government, mostly British and American English. It is political talk, not kitchen talk. Though actually, companies sometimes stretch it too: they want staff on-message with the brand. Same idea — stick to the script.