How to politely describe a situation where you have no publication because both you and the advisor are inexperienced in a new research topic?
I suspect the answer for this strongly depends on cultural context.  My answer is for applications to PhD programs in the US (and probably Canada also), but it may not hold for other countries.

In the US, it's not expected that every student going into a PhD program has written a paper.  (In some subjects, almost no students entering a PhD program has written a paper.)  Your advisor should simply not mention that you haven't written a paper, but explain what you have done successfully (and there are other notions of success than writing a paper!), and what positive character attributes you have that lead him to believe you will be a successful PhD student.  It will be necessary to go into details of your project, but no mention needs to be made about the lack of a paper coming out.

You should have a similar approach in your personal statement, emphasizing what you learned from your research project and your coursework and explaining what research you want to work on for your PhD.

The PhD program will be admitting you as a person, not your package of previous accomplishments (although of course your previous accomplishments will tell them something about you as a person).