According to G F Black's The Surnames of Scotland says that the name is most probably from Low German or Dutch groot meaning large or tall.
There's a story, which may or may not be true, that the place name John o'Groats is from Jan de Groot, supposedly a Dutchman who settled there. Wikipedia says, "The settlement takes its name from Jan de Groot, a Dutchman who once plied a ferry from the Scottish mainland to Orkney, which had recently been acquired from Norway by King James IV. Local legend has that the o' Groats refers to John's charge of one groat for use of his ferry, but it actually derives from the Dutch de groot, meaning 'the large'."
The earliest record of the surname is a charter in 1496 (during the reign of King James IV) by William de St Clair, earl of Caithness, of one pennyland in Duncansby, Caithess to John Grot, son of Hugh Grot.
