"Give me an apple, please."
Translation:Дайте мне яблоко, пожалуйста.
So яблоко is unchanged because it's neuter, or is this not a transitive verb?
Я is used for the subject of the sentence. Someone who is doing the action. So, «Я даю» means 'I’m giving, I give’. This form is called nominative case.
Мне is used for an indirect object, someone who is affected by the action, but not directly. In «Дайте мне яблоко» 'Give me an apple', яблоко is a direct object, because the ‘giving’ affects apple the most (it’s being moved around), and мне is the indirect object, becuase I am affected by ‘giving’ (I get one apple), but not to such an extent as apple is. Мне is the dative case form.
Russian has more cases, and you’ll learn them and their usage as you progress through the course.