How did cats and brooms become associated with witches on Halloween?

Witches of the middle ages were more often than not thought to be women, and a disproportionate number of those condemned as witches were in fact female.

Women of the time did the work of running the household and part of that included the cleaning.

Brooms were part and parcel of their "equipment" so when looking for "proof" that a woman was a witch, the accusers could point to the broom and say, "There is proof".

This dose not mean modern day witches don't use brooms (we call them Bessoms) as part of some of our Rituals. The Bessom is used to ritually cleanse the area we use for our Circle (Sacred space) and as a common part of Hand-fasting Rituals (weddings).