The pronoun "me" correctly completes the clause and shows that the teacher gave Fran a better grade than he gave the speaker.
The pronoun has to be a first-person one since it refers to the speaker - the one "talking". That's why the pronoun "she" would be eliminated from our list of options right away.
The reason why we cannot use the pronoun "I" is because it would change the meaning of the sentence. "I" is a subject pronoun while "me" is an object one. If we were to use "I", it would sound as if the speaker is the subject and, thus, is the one giving Fran the grade.
By using "me", the speaker makes it clear that he/she received a grade from the teacher - the speaker is the object of the verb "gave". And that is what the exercise is asking us to do: to complete the sentence so that the speaker has received the grade.
