'Threads' is a noun used for the raised part, circling part of the screw. The thread goes around the screw in a spiral pattern, and is what causes the screw to go in or out when it is turned.
'Hairline' is more commonly used to describe a crack in an object like a plate, that is very thin, almost invisible, and so might be mistaken for a hair resting on it's surface.
It is not a common combination. 'Hairline threads' simply means that the threads are very fine, perhaps delicate, perhaps thin.