How did the idea of witches riding broomsticks begin?

Answer 1

The stereotype of a "witch" on a broom is not a realistic one. Levitation is not easy, and not common.

There are a couple of possible explanations for the misconception however.

Witches at one time were thought to be women, often herbal healers who lived without the support/protection of a husband (either widows, or spinsters). In the times that these ideas formed a woman was closely associated with various forms of housework (sweeping - therefore brooms became one of the symbols). It is possible that there was a minor truth in this, as many practitioners of magic used (and still do) a staff or wand to channel power... it is easy enough to disguise a staff in a broom.

The idea of "flying" may have its origin in the practice of astral projection. The ability to "send" your astral self (kind of like a soul) elsewhere, either on this plain or on higher plains. There are old recipes for "flying ointment" that include hallucinogens, and mind altering plant substances.

Modern Witches do not actually "fly" on brooms, but some of us do practice astral projection, without the "flying ointment".

Answer 2

The stereotype of a witch in a pointed hat, riding on a broom stick has been with us for centuries. The connection with witches and brooms started with the old belief that witches were all women, and in the middle ages women meant "housework" hence the broom. There are many practitioners of witchcraft today that use a staff as a focus, this sort of thing can be (and was) easily disguised as a broom, and in the "burning times" subterfuge was not only commonly practiced, but also advisable.
The rumours of witches flying are most likely derived from the misinterpretation of the descriptions of astral travel. Many of the suppose flying ointments of the time included poisonous and or hallucinogenic plants and materials, increasing the misunderstandings.