I actually don't know what a "bonnet" is -- I think the Brits might use this more than we do in the U.S. Some type of hat, I think.
But there is an expression: "to have a bee in one's bonnet", which means to be excited, worried, or to cause a fuss (to cause excitement), to be angry. And, from the context of the passage above, I'd say the meaning is the same.
"Take it easy. Don't let him git under them bonnets." = Be calm. Don't let him excite you. Don't let him worry you. Don't let him scare you.