As has already been pointed out earlier in this topic, the issue is either a repeated phishing attempt (in which someone sends you a faked e-mail, directing you to a faked site, which tries to get you to "log in" so that the hackers can get your ID and password) or someone is repeatedly trying to get access to your account. I'm guessing the latter is more likely... someone's repeatedly trying to reset your password, for who knows what reason, and they're trying repeatedly because they're never getting the password reset e-mail. (Why they would expect to receive that e-mail I don't know... people are stupid.)
Make sure your Apple ID has a good, secure password. (There is no need to reset your own password to do this, just go to appleid.apple.com and change it.) I'd also advise turning on two-factor authentication to further protect your Apple ID:
Apple introduces two-factor authentication
This will prevent password reset attempts, as it will not be possible to even start the password reset process without having access to the device used to receive the code used as the second identification factor.
Note that getting rid of your iPhone and switching to Android over this would be quite silly. Most likely, the person doing this knows you, and might try the same thing on some other account. Besides which, you'd be switching a very secure phone for one running the least secure system on the planet. At this point, Android has surpassed Windows in terms of malware "market share."