There are a few companies pitching this same cryogenics nonsense for high
accuracy target rifle barrels. The reviews (based on comparative testing)
I've seen say it doesn't hurt, but it doesn't result in any great
improvement, either.
The advertising words you have quoted are simply gibberish. Phrases such as
"residual stress is outward, adding force on the ball" don't witstand
scrutiny. A residual stress can't be "outward" (it's either tensile, shear
or compressive, and must be balanced by a similar yet opposite amount of
stress elsewhere, or the bat would be undergoing deformation). What force
would be causing such a stress? It is as though this cryogenic treatment is
being pitched as a sort of reverse annealing process.
Besides, isn't your husband's purpose in playing baseball to have an excuse
to go out and drink a few beers with his buddies? Tell him to lighten up
and enjoy himself. Spend the money on something you'll use together.