The image is of clothing drying on a clothesline. It dries all by itself with no help from anybody. When you "hang somebody out to dry", you are severing your connection with them, allowing them to recover, or not, on their own.

If I hang someone out to dry, I leave them in a bad situation when you could potentially help them. There are many possible situations where we would use this, but perhaps most commonly we use this when someone's trust is betrayed. For example, I ate the last piece of cheesecake, secretly. Only my sister knew. And then later on, when everyone was looking for the cheesecake, my sister told them that I ate it. "Way to hang me out to dry!" I said. So now, I am hung out to dry—the bad situation for me is that everyone can blame me for eating all the cheesecake. I would say this is American English.

I also agree that it is American English which means to get someone into trouble, especially by making them take the blame for a bad situation.