In a babyish or immature manner; in the way a baby acts or speaks.
From 'babyish' + the adverbial suffix '-ly.' 'Babyish' comes from 'baby' + '-ish' (somewhat resembling). The '-ly' converts the adjective into an adverb describing how an action is performed.
The word 'babbishly' is almost never used—'childishly' sounds better to modern ears—but it's a reminder that English once had more '-ish' adjectives that could be turned into '-ly' adverbs, a system we've mostly abandoned.
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