I am not sure how those guys did it, but I was talking to a friend of mine today, and I guess that it had something to do with forcing the server to use its private key to check for information sent to it. Then you use the heartbleed bug to intercept the intermediate forms on the information you sent to be decrypted/authenticated. Since you know the plaintext, the ciphertext and the intermediate forms, it should be possible to recover the key.

As I said, I am not sure that is right or if that was the method used to exploit cloudflare, as I didn't had the time nor the knowledge of openssl implementation to test it out, I am just throwing my guess out there before the official exploit comes about.

edit: formatting