Learn Farsi: امروز کافه شلوغه، بیا بریم ویلا! - The cafe is crowded today, let's go to the villa!

In written Farsi, you would indeed write برویم. But in colloquial Tehrani, the spoken stem of رفتن (to go) shrinks. Just as the present tense می‌روم becomes میرم, the subjunctive برویم drops the و and becomes بریم.

Why is the word بیا (come) used here if the translation says let's go?

In Farsi, the imperative بیا is frequently used as a prompt meaning "come on" or "let's" to encourage someone to do something with you. When used this way, it is immediately followed by the main verb in the subjunctive mood, which is بریم in this sentence.

What is the ه at the end of شلوغه doing?

This is a very common spoken contraction. The formal written phrase is شلوغ است (is crowded). In colloquial Farsi, the verb است (is) loses its first letters and attaches to the previous word as an "e" sound, which is written as a silent ه.

Why isn't there a preposition like به (to) before ویلا?

While you would write به ویلا in formal texts, spoken Farsi almost always drops the preposition به when expressing motion toward a destination. You simply say the verb of motion followed directly by the place.