To explain why, we need to go back to Grammar 101 and the parts of speech. In the phrase above, "between" is a preposition. A preposition gives information about location, space, direction, and time. The noun or pronoun to the right (typically) of the preposition completes its meaning. It's called the object of the preposition and is always in the objective case. In English, case changes only in pronouns, not nouns. The following personal pronouns are in the objective case: me, you, her, him, it, us, and them. Since "between" is a preposition, its pronoun has to be in the objective case, so "me" rather than "I" is correct. Let's look at some other examples of prepositions and their objects.
- Stay with me.
- Water dripped from the pipes above them.
- The tickets were for us.
- The bird flew toward her and me.
- Stay with I.
- Water dripped from the pipes above they.
- The tickets were for we.
If you remember that prepositions take pronouns in the objective case and to swap the pronoun order when in doubt over "I" and "me," you won't go wrong in your writing. And don't worry. You don't have to keep this information just between us.