Learn Russian: Здесь вещи.

The singular word is вещь (thing), which ends in a soft sign (-ь). To make these nouns plural, we replace the -ь with -и. Also, because of a strict Russian spelling rule, you can never write the letter ы after щ anyway!

Where is the word for "are" in this sentence?

Russian skips verbs like "am," "is," and "are" in the present tense. Здесь вещи literally means "Here things," but it is the perfectly correct way to say "Here are the things."

How do we know it means "the things" instead of just "things"?

Russian does not have articles like "a" or "the." You just use the noun by itself, and the listener figures it out from the context.

Could I use вот instead of здесь?

You could, but it changes the meaning slightly. Здесь is about location (the things are physically located in this spot). Вот is used when you are pointing at something or presenting it, like saying "Here they are!"