Learn Icelandic: Langar þig að fara heim?

This is exactly what a 'quirky subject' is! The verb langa (to want) requires its subject to be in the accusative case, not the usual nominative case. So instead of þú (you), we use the accusative form þig. Literally, it is a bit like saying 'It longs you'.

Why does the sentence start with Langar?

In Icelandic, to make a yes/no question, you swap the verb and the subject. In a statement, it would be Þig langar... (You want...). By putting the verb first, it becomes a question: Langar þig...? (Do you want...?).

Does langar change if I am asking a group of people?

No, it stays exactly the same! With quirky subjects, the verb is always locked in the third-person singular form (langar). Even if you ask a group using the plural pronoun ykkur, it will still be Langar ykkur að fara heim?.

Why use heim instead of heima?

Heim shows movement toward a place (going home). If you were talking about being at a location without moving, like staying at home, you would use heima. Since the verb fara (to go) involves movement, we must use heim.