Learn Italian: Devi lavare i piatti. - You have to wash the dishes.

Because it follows the modal verb dovere (here conjugated as devi). When you use a modal verb like dovere (to have to), the main action verb that follows it always stays in the infinitive (the dictionary form ending in -are, -ere, or -ire).

Why isn't there a word for 'to' between devi and lavare?

English says 'you have to wash', but Italian modal verbs connect directly to the next verb without any linking words. Plus, the infinitive lavare already means 'to wash' all on its own!

Does piatti mean 'plates' or 'dishes'?

It means both! While a single piatto is a plate, the plural i piatti is the everyday term used for 'the dishes' when you are talking about washing up.

Why isn't the word tu (you) at the beginning of the sentence?

Italian usually drops subject pronouns because the verb ending gives it away. The form devi specifically means 'you have to', so adding tu isn't necessary unless you really want to emphasize who has to do the washing.