They are called Rhumb Lines. Their purpose is to aid in navigation.
As far as I understand, which is to say I'm probably incorrect, navigators would align their ships with that point on a map then steer on that compass point until the next node whereupon they would alter their course accordingly. Note that they probably also used parallel rulers so they didn't actually have to be at the node physically. This would allow them to navigate quite easily without having to account for projections etc.