This is another piece from Hobgoblin & Sweet Puck Fairy Names & Natures by Gillian Edwards. It seems to explain the origin of riding a broomstick.
The fairies did occasionally ride horses, either their own or those they stole from mortals, or make themselves mounts out of straws from the fields, but chiefly as a pastime or for their amusement. ‘When fairies remove from place to place’, explained John Aubrey, ‘they are said to use the words Horse and Hattock.’ No sooner had they pronounced the magic words than they were where they wished to be; ‘their chamaelion-lyke Bodies swim in the air near the Earth with Bag and Bagadge,’ adds Kirk,… Hattock is good Scots and means a little hat; the phrase might therefore be equated with our own ‘Boot and saddle’ except that this is said to represent the French cavalry order Boute selle, put on the saddle, and has no relation to boots.
Isabel Gowdie, a self-confessed Scottish witch, told a court in 1662 that witches as well as fairies ride on straws and in whirls of dust that sometimes blow up on a dry road. “We will flie lyk strawes quhan we pleas; wild-strawes and corn-strawes will be horses to us, and we put thaim betwixt ur foot, and say, Horse and hattock, in the Divillies name!’ So if you see straws in a whirlwind it is safest to cross yourself, raise your hat and speak politely; for those who do not, ‘we may shoot them dead at our pleasour. An that ar shot be us, their sowell will go to Hevin, bot their bodies remains with us and will flie as horsis to us, als small as strawes.’