"Feeling his oats" is a saying we use when talking about a horse that has far too much energy, is struggling to focus and is more inclined to bounce about fidget, or run off, than walk.
Is there any truth in oats (or other grains) being responsible for making horses "hyper"?
To a point, yes.
We feed grain to give horses extra concentrated energy so they can do more work. (Many pleasure horses are perfectly fine with little or no grain).
If your horse has a lot more energy than he needs for the work he's doing - then he'll become hot and hyper. (Horses may also become hot or hyper in cold weather just because they want to move around and get warm). It doesn't matter what the food is - in fact, corn and barley have more concentrated energy than oats.
So, hyperactivity is not a symptom of a particular kind of food - it's a symptom of too much grain period.