One understated reason that it gained momentum was that it was an outlet for popular nativist sentiments by WASPs (especially among the middle class) against the wave of newly arrived immigrants from cultures where social drinking was more common. Some Progressives at the time viewed the elimination of social drinking through Prohibition and the closing of the Saloons associated with immigrant drinking as a step towards assimilating the urban immigrant population into WASP social norms. Some among the Progressives spoke of this culture in medical or even eugenic terms as some contagion that was associated with the immigrants that must be stopped to prevent some sort of "degeneration" of society that would come with the adoption of drinking and drinking culture by WASP America. These reactionary sentiments would see their apogee in the 1920s, where there was an alliance in the South and rural Midwest between the KKK and local Anti Saloon League chapters to enforce local Prohibition.

Prohibitionist anti immigrant attitudes also got a jolt of support when America entered WWI and the beer-distilling sections of the German-American community became a target for pro-war sentiments.

Though I must say that anti immigrant sentiment was only one facet of the many intersecting prejudices of early 20th century America that influenced the law's adoption. You could also see cause from a desire to reinforce patriarchal social relations, strengthen existing regimes of racial segregation, or even as a way that capital sought to impose standards of discipline on labor (see Henry Ford's Sociological Department). Basically (as others have stated) many of these reasons boil down to the reactionary ideology of middle class America.