How to withdraw from an application process in a polite manner?
Just send a polite email stating that you found another position or that you simply prefer to take a different direction. Be sure to thank the professor for the time he has put into you. The email does not have to be long, but it should give a reason. You do not need to be overly elaborate, though. And don't forget to be polite.

The thing you do not realise is that the professor wants to find a student who will be committed to his topic for several years to come. If you are not such a student, it is best for the professor to know that before taking you on as a student. In a sense, you are doing the professor a favour by withdrawing.