When something is not very probable, but you did it anyways... HOW ON EARTH DID YOU DO IT??? what were the chances for this to happen? If you delete "on earth" you will have a common, normal daily expression... but if you add "ON EARTH" you are giving to the question the surprise and "oh God, I don't believe it " factor . ^^

I think maybe it is how could you know who are you and what you should do,or what can you contribute to our home -earth.

It just means: "How did you do it?" The "on Earth" is used for emphasis, perhaps to show surprise or disbelief on the part of the asker.

I think it's referring to something that a person didn't know could happen. I believe the "on earth" is just used to exaggerate that surprise.

the Chinese meaning of "on earth" here is: "究竟"“到底”。整个句子：你到底怎么做的？