A "mouthpiece" is a person who speaks for someone else. A "stooge" is someone who does another person's "dirty work."

To be a "mouthpiece" or a "stooge" in this context means to naively reiterate what the source told you, without further investigation. It means to take what the source said, believe it, and publish it as a story, without doing the investigative journalism necessary to find out if the statement is true. Many times people who are being interviewed by journalists want to be talked about in the news story in a way that will make them look good, or will further their agenda, and so they will twist their words or leave out the details that would cause them to look bad or will hurt their agenda. A journalist who does not do enough investigation, and publishes only the details that the source wants to be published, is operating as a "mouthpiece" of the source. Think of it as the source is using the journalist as a physical extension of the source's mouth, to communicate only what they wish to communicate to the public.