What does it do? I see items in game that raise and lower it, but what effect does it have being high or low?

Sorry this must seem like a total newb question, but the only thing I could think was making the monster more responsive to directions on the touch screen but... not that I could see.

Anyone know?

Nobody knows?

Higher Wits means higher success rate in training. The guide essentially says -

High Wits give:
Higher success in Training
Better results in Drills and Errantry
Better combining.

Low Wits give:
More Failures and Cheating in training

That's interesting. One wonders why you would ever give the items designed to lower Wits. Anyway, thanks!

You don't want the Wits stat to go to max. The monster then acts like a know-it-all and actually cheats more.