Why did France and Spain become increasingly interested in the American Revolution?
As a means of limiting the power of the English empire.
France had lost the 7 year global war with England. This conflict also know as the French and Indian wars had started in the Ohio Valley of the Americas. As a result of the war French had lost its control of north America and the fur trade. England had also gained preeminence in Indian and the south Pacific.
The French saw the American Revolution as a means of hurting the English and reducing the power of the English Empire. Reducing their enemy England would increase the power of French Empire.
Spain feared English encroachment on their possessions in America. Spain claimed what is now Florida, Mississippi, and the Southwest states of Texas, New Mexico, Arizona, California and Nevada. Also English privateers ( pirates) had preyed on Spanish shipping in the Americas. Spain hoped that the America revolution would ensure the safety of their shipping and possessions from the English.
Both Spain and France had an interest in the success of the America Revolution as a means of weakening England.