Hi!
Well, you need to think in Japanese.
I think that the meaning of the kanji 方 has to do a lot with the Japanese way of thinking. Side, direction, way, person (politely) ... and, in my opinion, addressing to, sense, about something...
...の方が... is just a set phrase that mean "more than" as you wrote.
I don't know how to translate it but I try to understand just the concept trying to think in Japanese.
Maybe, The Rosetta Stone method is not a good resort to this kind of problems.
I try to translate the way of thinking instead the word.
That's an unavoidable problem about learning languages.