"starts" is a manufacturing term. It is a noun that refers to how many orders have been placed, which leads to how much of the product will "start" the production cycle.
"Housing starts" refers to how many new homes are being contructed.
Housing starts fell means that there are fewer houses being started. Housing starts fell more sharply than expected = the number of new houses being started was a lot lower than people had thought. They fell to such a low number that this number was the lowest in more than three years. The original sentence was correct. (It does not refer to prices, either.)
Any type of manufacturing can have starts. When I was at a manufacturing plant, "starts" was one of the most watched numbers. It told us how busy we were going to be.
Does that help?