In an elegant, stylish, or refined manner; done with sophistication and good taste.
From 'classy' (elegant, high-class) plus the adverbial suffix '-ly.' Informal usage that became common in 20th century English.
The word 'classily' is delightfully slangy for something supposedly elegant—it's like saying someone dressed 'fancily,' which is exactly what non-snobby people would say.
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