मुझे is the oblique form of मैं . The oblique pronouns are listed in the lesson notes here: https://www.duolingo.com/skill/hi/Food/tips-and-notes. They go on to say "However, these are used in few categories of sentences and find no substance in this lesson other than being mentioned", but that is evidently untrue haha. Oblique pronouns are all over the sentences in this lesson, they basically translate as "to me", "to you", "to him", etc.
So मुझे और चावल चाहिए literally means "to me more rice is wanted".
A more English-similar sentence using मुझे is: वह मुझे किताब देता है which means "he gives the book to me".