Learn French: Le serveur travaille plus vite.

Vite is an adverb, meaning it describes how an action is done. Here, it describes how the waiter works (travaille). Rapide is an adjective, which describes nouns (like the waiter himself: Le serveur est rapide). To say "works faster" (literally "works more quickly"), we must use plus vite.

Since we are learning about intensifiers, could I say beaucoup vite to mean "very fast"?

No, you cannot use beaucoup (a lot / much) right before an adjective or an adverb like vite. To say he works "very fast", you would use très and say Il travaille très vite. In this sentence, plus is used to mean "more fast" (faster).

Can I use beaucoup with the verb travailler in other sentences?

Yes! If you want to say the waiter "works a lot" (intensifying how much he works, rather than his speed), you would say Le serveur travaille beaucoup. But because this sentence focuses on his speed (vite), we use plus directly in front of the adverb.

Do I pronounce the 's' at the end of plus in this sentence?

No, the 's' is silent here. When plus means "more" and is followed by a word starting with a consonant (like vite), you do not pronounce the 's'. It sounds like "plu".