The strict origin of a golf links is that it was built on land reclaimed from the sea and as Clarion St says, it forms a link between the land and the sea. Strangely in the very early 1900s, Sidcup Golf Course in North Kent which was 18 holes in those days was described in a newspaper report of the day that 'it is one of the finest golf links in the south of England'. Hardly correct as it is about 50 miles from the nearest sea.
One further point is that a 'golf links' is deemed to provide what is considered to be the truest test of golf and which endures extreme changes in weather conditions. Hence The Open Golf Championship has to be played on a links course.