Learn Polish: Żeby zapobiegać chorobom, trzeba myć ręce. - To prevent diseases, one needs to wash hands.

The verb zapobiegać (to prevent) always takes the Dative case. Think of it as acting in opposition to something, very similar to the logic of the preposition przeciwko (against) which you are learning in this lesson and which also requires the Dative. Here, chorobom is the Dative plural of choroba (disease).

Why do we use the infinitive zapobiegać immediately after żeby?

When żeby means 'in order to' and the subject of the sentence doesn't change (or in this case, applies to a general 'one'), it is followed directly by an infinitive. This is the standard way to express purpose or intention in Polish.

Could I use a form of musieć (must) instead of trzeba?

You could say something like musimy myć ręce (we must wash our hands), but trzeba is a better fit here. It is an impersonal verb meaning 'one needs to' or 'it is necessary'. Since this sentence is offering general health advice meant for everyone, an impersonal structure sounds the most natural.

Why is the word for 'hands' just ręce without a possessive pronoun like 'swoje' (one's own)?

In Polish, you generally do not use possessive pronouns with body parts or clothing if context makes it obvious whose they are. Saying myć swoje ręce (to wash one's own hands) sounds strangely emphatic, as if you need to clarify you aren't washing someone else's hands!