Learn Italian: Torno a casa per mangiare con la mia famiglia.

In Italian, the verb ending usually tells you who is doing the action. Because torno ends in -o, it already means "I return". You don't need to add the word for "I" (io) unless you want to add special emphasis.

Why is it a casa instead of alla casa (to the home)?

The phrase a casa is a fixed expression in Italian, just like "going home" or "at home" in English. You drop the word for "the" (la) when talking about going to your own house.

Doesn't per mean "for"? Why is it used here as "to eat"?

While per often translates to "for", when you place it directly before an action verb like mangiare (to eat), it means "in order to". So per mangiare translates directly to "in order to eat".

Why do we say la mia famiglia? I thought we don't use "the" with family members.

It is true that you drop the article for specific singular relatives, like saying mio padre (my father) instead of il mio padre. However, the word famiglia (family) is a group noun rather than a single relative, so it keeps the article la.