Learn Spanish (Latin America): Llueve, así que el mensajero deja el paquete sin hablar. - It is raining, so the messenger leaves the package without speaking.

We use sin + infinitive (hablar) because the subject doesn't change. The messenger is doing both actions: leaving the package and not speaking. If you wanted to say he leaves it without you speaking, the subject would change, so you would need sin que + the subjunctive: deja el paquete sin que tú hables.

The English translation says "It is raining," but llueve is the simple present. Shouldn't it be está lloviendo?

You can definitely say está lloviendo. However, in Spanish, the simple present (llueve) is used much more often than in English to describe what is happening right at this very moment. English relies heavily on the continuous "-ing" form, but Spanish often prefers the simple present tense.

Is así que the most common way to say "so" in this situation?

Yes, así que is the most natural, everyday way to say "so" or "therefore" when connecting a cause to its result. You will also hear entonces (so / then) used a lot in spoken Spanish, or por lo tanto (therefore) in slightly more formal situations.

I learned that dejar means "to let" or "to allow." How do I know it means "leaves" here?