"the hell" is an intensifier. It makes the remark stronger.
"What would I know" becomes more emphatic by saying "What the hell would I know?"
It's like saying, "What on earth would I know?" or "What in heaven's name would I know?" These are all intensifiers.
Since "hell" is a very mild oath, it is the least polite of these forms. A sweet little old lady would not use this expression at a tea party.