Learn Finnish: Syö voileipä ja juo mehu! - Eat your sandwich and drink your juice!

This is the main rule of this lesson! Because the verbs syö and juo are in the imperative (command) form, their total objects drop the usual -n ending. They look exactly like the basic nominative dictionary form, even though they are the direct objects of the sentence.

I thought we were supposed to use the partitive case for food and drinks. Why isn't it Syö voileipää ja juo mehua?

You use the partitive when you mean 'some' of something. If you used the partitive here, it would mean 'Eat some of the sandwich and drink some juice'. By using the total object form instead, the command is to finish the whole sandwich and the entire portion of juice.

How did we form the command forms syö and juo?

Both syödä (to eat) and juoda (to drink) belong to verb type 2. To get the informal command form (imperative), you simply take the dictionary form and remove the final -dä or -da.

The English translation says 'your sandwich' and 'your juice'. Why aren't there any words for 'your' in the Finnish sentence?

In Finnish, it is very common to leave out the words for 'your' when the context makes it obvious, especially in simple daily commands. Adding sinun (your) and the possessive endings would sound unnecessarily formal here.