The Widen upgrade is used to make your regular Broom attack much more effective. Instead of attacking just one space ahead, harmful dust will be emitted on each side of the broom, resulting in a much wider area of damage.
A few words on the broom extension read "Equip the WIDEN upgrade in the menu to have the broom release harmful dust to the left and right."
This upgrade can be found in one of the caves in the Red Sea. The entrance is the southernmost cave in the Red Sea area. Technically, it is located in a part of the Red Cave, although it cannot be accessed from the normal entrances.