Actually if you think about if two people were driving their cars somewhere and at the beginning of the trip, one person turns to the other and says "Are you following me or am I following you?"
Despite the fact that this sentence is clearly present tense in English, the speaker is indicating the future and "will you follow me or will I follow you?" holds borderline the same intent as the former sentence. So this is a preculiar feature of both German AND English :)
The verb 'folgen' is a rare exception. Most verbs are accusative, but this one is dative. 'Mich' is accusative, 'mir' is dative. There's no good explanation, it's a total exception. You just have to remember that it's dative, and thus the 'me' in question is 'mir'.