Learn German: Wir kochen zusammen. - We are cooking together.

Because the subject is wir (we). For regular verbs, the wir form looks exactly like the dictionary form (the infinitive), which ends in -en.

The translation says "We are cooking," but where is the word for "are"?

German doesn't have a separate "-ing" tense (present continuous) like English does. The single phrase wir kochen translates to both "we cook" and "we are cooking". You never need an extra word for "are" when describing what you are doing right now.

How would this sentence change to say "You guys are cooking together"?

You would use the plural "you" pronoun, ihr. The regular verb ending for ihr is -t, so you would drop the -en and add -t. The sentence becomes: Ihr kocht zusammen.

Does kochen only mean "to cook"?

It usually means cooking a meal, but it also translates to "to boil". For example, you use it to say that water for pasta is boiling. In this sentence, with zusammen (together), it clearly means cooking a meal.