What does it mean when a consultant for the competing station asks you for a tape?
It means the consultant thinks you're talented. An old consultant tactic to help the client is to find jobs for the good people who work for the competition. They may not improve their own client's station, but through addition by subtraction, they are making the competition weaker by moving the good people.
But don't confuse them with agents. They're not going to push you as an agent would, but they will put you in their database of talent. For instance, if I'm a client with a consulting firm and I need an anchor, I can simply call the consultant and say, "Send me all your female anchors," or "send me all your meteorologists" and I'll get a tape with a bunch of people on it.
People do get jobs this way, but it doesn't happen too often. Consider it a compliment, but don't slack off in your own job hunting tactics.