Learn Italian: Tutta la spazzatura, la dovete differenziare.

This is a structure called left dislocation. When you move the direct object (Tutta la spazzatura) to the very beginning of the sentence to establish it as the main topic, Italian requires you to refer back to it with a clitic pronoun. Since spazzatura is feminine singular, we use the matching pronoun la right before the verb.

Could I attach the la to the end of differenziare instead?

Yes! Because dovere is a modal verb (a helper verb like can, must, or want) followed by an infinitive, you have two choices for where the pronoun goes. You can put it before the conjugated verb (la dovete differenziare) or drop the final -e of the infinitive and attach it there (dovete differenziarla). Both mean exactly the same thing.

What does differenziare mean here? Doesn't it usually mean to differentiate?

Literally, yes. However, in the context of household waste, differenziare is the standard verb for sorting or separating trash for recycling. You will also frequently see the noun phrase la raccolta differenziata, which means waste sorting or recycling.

Why do we need the article la in Tutta la spazzatura?