How did the idea of witches riding broomsticks begin?

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".