Learn Portuguese (Portugal): Para de falar, por favor!

You are thinking of the preposition para, but here para is actually a verb! It is the informal command (imperative) form of parar (to stop). Because the command form of the verb and the preposition look and sound exactly the same, you have to rely on context to tell them apart.

Why do we need the word de in this sentence?

In Portuguese, the verb parar requires the preposition de when it is followed by another action. The set pattern is always parar de + [infinitive verb], which means 'to stop doing [something]'.

The translation says 'talking', so why use falar (to talk) instead of an -ing form?

In English, we use 'stop' followed by an -ing word (stop talking). However, in Portuguese, whenever a verb comes directly after a preposition like de, it must stay in its basic, unconjugated infinitive form. So, 'stop talking' translates literally as 'stop of to talk'.

Who is being told to stop talking here? Is it formal or informal?

This is an informal command directed at a single person you know well, corresponding to the tu form. If you wanted to say this formally to someone (using você), the command form would change to pare de falar.