The chaur sahib is a whisk of fine hair on a handle waved over the Guru Granth Sahib whenever it is open, and its meaning comes entirely from the standing of the scripture as the living Guru.
Honour to a sovereign
The origin of the object is entirely secular: the whisk, waved to keep insects from a person of high rank, was among the marks of royalty and honour in the courts of the region, and its use over the scripture treats the Guru Granth Sahib with the honour due to a living sovereign. An attendant waves the chaur over the open volume as a continual act of service and reverence, so the object expresses in gesture the Sikh conviction that the scripture is not a book about the Guru but the Guru present.
The scripture as Guru
The whole treatment of the Guru Granth Sahib, the canopy over it, the cloths that wrap it, the raised seat, the waving of the chaur, follows from the declaration of Guru Gobind Singh that after him the Guruship would rest in the scripture, so that the volume is attended as a living teacher and sovereign would be. The chaur is one element of that ceremonial honour, and its royal origin makes the point sharply: the marks once reserved for a king are given to the book because the book holds the authority the Gurus held. To wave the chaur is thus to serve the Guru, and the object belongs to the court of honour built around the scripture.