Answer by taking a quote from the following article:

Cullen was born in Hamilton, Lanarkshire. His father William was a lawyer retained by the Duke of Hamilton as factor, and his mother was Elizabeth Roberton of Whistlebury. He studied at the Old Grammar School of Hamilton (renamed in 1848 The Hamilton Academy), then, in 1726, began a General Studies arts course at the University of Glasgow. He began his medical training as apprentice to John Paisley, a Glasgow apothecary surgeon, then spent 1729 as surgeon on a merchant vessel trading between London and the West Indies.

In 1755 he was enticed by Lord Kames to become Professor of Chemistry and Medicine at the University of Edinburgh. It was in Edinburgh, in 1756, that he gave the first documented public demonstration of artificial refrigeration. Cullen used a pump to create a partial vacuum over a container of diethyl ether, which then boiled, absorbing heat from the surroundings. This created a small amount of ice, but the process found no commercial application.  From 1757 he delivered lectures on clinical medicine in the Edinburgh Royal Infirmary. On the death of Charles Alston in 1760, Cullen at the request of the students undertook to finish his course of lectures on materia medica; he delivered an entirely new course, notes of which were published in an unauthorised edition in 1771, but which he re-wrote and issued as A Treatise on Materia Medica in 1789.  On the death of Robert Whytt, the professor of the institutes of medicine, Cullen accepted the chair, at the same time resigning that of chemistry. In the same year he had been an unsuccessful candidate for the professorship of the practice of physic (medicine), but subsequently an arrangement was made between him and John Gregory, the successful candidate, by which they both agreed to deliver alternate courses on the theory and practice of medicine. This arrangement continued until the sudden death of Gregory in 1773. Cullen was then appointed sole professor of the practice of physic, and he continued in this office until a few months before his death.  In 1783 Cullen (together with his sons) was a co-founder of the Royal Society of Edinburgh.  He died on 5 February 1790 in Kirknewton, West Lothian, and is buried in the churchyard there. His son Robert lies with him.

Why did he go there?
he was enticed by Lord Kames to become Professor of Chemistry and Medicine at the University of Edinburgh.