To indicate the reflexivity of the verb the "myself" is required. Reflexivity means that the subject and object of the verb are the same.
If you use "me", the subject and object of the verb will be assumed to be different. The sentence "Look at me" is requesting that the listener look at your image in the mirror. In "I look at me in the mirror" there is a conflict with the established subject of the sentence "I look at . . ." and the expected change of object to someone else. Therefore it is grammatically incorrect and confusing to the listener.