In a fashionable, elegant, or stylish manner; the way a dandy would dress or behave.
From dandy (the noun/adjective) + -ly (the suffix that turns adjectives into adverbs). This became the standard English adverbial form by the 19th century, replacing earlier variants like dandilly.
Dandily is actually rare in modern English—you almost never hear people say it—but it perfectly shows how you can predict English grammar: take 'dandy' and add '-ly' and boom, you've got the adverb! Most English speakers who knew what a dandy was understood this rule instantly.
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