What's the "real" reason that IP=PSPACE is non-relativizing?
Any answer to a question of the form, "What is the real reason that..." will necessarily be somewhat subjective.  However, for the particular case of IP = PSPACE, I think that a pretty good case can be made that arithmetization is indeed the key, by observing that while IP = PSPACE does not relativize, it does algebrize in the sense of Aaronson and Wigderson.  As they explain in their paper, roughly speaking, a complexity class inclusion ${\cal C} \subseteq {\cal D}$ algebrizes if ${\cal C}^A \subseteq {\cal D}^{\tilde A}$ for all oracles $A$ and all low-degree extensions $\tilde A$ of $A$.  In particular, they show that the inclusion PSPACE $\subseteq$ IP algebrizes, even though it doesn't relativize.


  The intuition that I've developed is that non-relativizing results are ones that poke around with the nitty gritty of Turing Machines


This is not a bad intuition, but I think that the Aaronson-Wigderson result shows that the IP = PSPACE proof pokes around in a rather limited way, and certainly not in a sophisticated enough way to prove P $\ne$ NP, since Aaronson and Wigderson also show that non-algebrizing techniques will be required to separate P from NP.