actually 'sano' is an adjective, not an adverb, it's called a 'predicative complement', and in this sentence it's referred to the subject ('il bambino'), not to the verb. In fact, in the plural form, you would say: 'i bambini crescono sani' not 'i bambini crescono sano'.
In Italian we have two main types of verb: predicative verbs and copulative verbs. Predicative verbs have their own meaning by themselves; for example vivere, correre, leggere, and so on. Copulative verbs, on the other hand, need a noun or an adjective to complete their meaning; for example essere, diventare, chiamare. You cannot say just 'Mario è diventato', but you need a noun or an adjective after it: 'Mario è diventato papà', 'Mario è diventato biondo'. The adjective or noun is called 'predicative complement'.
Some verbs, like 'crescere', 'vivere', 'morire' are usually predicative (you can say: 'il bambino cresce', 'ho vissuto', 'è morto' without any predicate) but sometimes they can be used in a copulative way: "il bambino cresce sano", "ho vissuto felice", "è morto povero". Note that the adjective is referred to the subject, not to the verb, because being healthy refers to the child not to his growing, and being poor refers to him, not to his death.
There is a subtle difference with the adverbial adjective that, instead, is referred to the verb to better explain is meaning; for example 'lavora duro', 'parla chiaro', 'mangio sano'; in these sentences the adjective is referred to the verb, not to the subject, therefore it has an adverbial role. In fact, if you turn the sentences to the plural form, the adjective doesn't change: 'lavorano duro', 'parlano chiaro', 'mangiano sano'.