"About" is provoking a broader response.
Sometimes it is absolutely needed. For example, "Tell me about yourself," or "tell me about your day." Neither "yourself" nor "your day" is a specific piece of information that you can ask for. You can only ask for information about those things.
Sometimes its is optional e.g., "Tell me about what happened," vs, "Tell me what happened." The meaning is similar but the first invites a more detailed response, whereas the second is more direct, and perhaps urgent in tone. In this instance we could even shorten it to "What happened?", which we couldn't do in the prior examples. Another example could be: " Tell me about why you did that ", " Tell me why you did that", "why did you do that?"
If you are asking for very specific, exact information, e.g: "Tell me the truth", " Tell me the time"; then "about" wouldn't be appropriate.