Learn Italian: Qui i bambini possono giocare liberamente.

In Italian, adverbs of manner are usually formed by taking the feminine singular form of an adjective and adding -mente. The adjective for free is libero. Its feminine singular form is libera, so we add -mente to get liberamente (freely).

Why is giocare left in its infinitive (dictionary) form here?

Because it follows possono, which is a form of the modal verb potere (to be able to / can). When you use a modal verb, the main action verb that comes directly after it must stay in the infinitive.

Why does liberamente come at the very end of the sentence?

Adverbs of manner in Italian generally follow the verb they describe. Here, liberamente describes how the children can play (giocare), so it naturally sits right after it at the end of the sentence.

Could I use giocare if the children were playing the piano?

No, giocare is only used for playing games, toys, or sports. If they were playing a musical instrument like the piano, you would use the verb suonare instead.