I would guess that it's to keep underage girls out of the stripping business.

Probably previously the onus was on the owner to avoid hiring underage girls. But a couple scandals later, the government responded by requiring licensing.

That and chemicals for dying, perming and straightening hair, which can have dangers to both customer and hairdresser if misused. On the other hand incumbents sometimes band together and demand regulations to keep out new entrants to a business, as in a recent controversy over whether a license should be required to braid hair (a popular service in the black community).