Learn German: Man braucht das Megafon und die Plakate, um laut zu protestieren. - One needs the megaphone and the posters in order to protest loudly.

It translates to "in order to". The word um starts the clause, and zu is placed immediately before the infinitive verb at the very end of the sentence (zu protestieren).

Why use braucht instead of a form of müssen here?

In German, brauchen is used when you need a noun (like a physical object such as the megaphone). müssen (must / have to) is a modal verb used when you need to do an action, which would require another verb instead of a direct noun object.

Why is the word for "loudly" just laut? Doesn't it need an ending like "-ly" in English?

In German, most adjectives and adverbs share the exact same form. You just use the base word laut for both "loud" and "loudly", without needing any special adverb endings.

I always mix up man and Mann. Which one is used here?

Here we use man (lowercase, one 'n'), which is an impersonal pronoun meaning "one", "people", or the general "you". By contrast, der Mann (capitalized, two 'n's) means "the man" or "the husband".