I was in the Kyoto conference and I'm starting to slowly understand
the material. I've been working with Hoshi, Yamashita and a few others
on a more accessible introduction. Stay tuned.

>Does this theory change the way you look at baby tier arithmetic problems?

No (at least not yet), in the same sense that topos theory doesn't change
the way you look at sets.

>Does it make it easier to understand some trivial concepts
solve some interesting class of problems we couldn't before?

I don't think it makes anything easier (at least at this point).
If anything, it makes certain things possible. We're currently working
on transporting some of the techniques to existing mathematical settings
so that a larger portion of the community can join, learn, use and research the theory.

By the way, the paper linked in OP, 'From Gaussian Integral to IUT' is really good.
The first chapter should be accessible to anyone
with a freshman-level Calc knowledge, the last chapter is also useful.
It gets a wee bit hairy in between, but experts in the field should be able to grasp that