A privy midden is like a fancy way of saying a toilet from a long long time ago. People used to use it before there were modern toilets. It was usually a small building outside of the main house that had a hole in the ground where people would do their business.
Afterwards, instead of flushing it away with water like we do today, people used to cover it up with dirt or sawdust to make it smell better and keep it from being messy. Over time, the waste would turn into compost, which is like really good soil that helps plants grow.
So basically, a privy midden was a way for people to go to the bathroom and then turn their waste into something useful for the earth.