The possessive form "my" is correct here, because "being" is a gerund (–ing form of a verb used as a noun). Technically, "me" is incorrect grammar, although it is commonly used.

I agree with Randy. 'Being' is a gerund form. A gerund, basically, is when a verb does the job of a noun. Nouns are modified with possessive adjectives: my, your and so on. You can analyse that part of the sentence like this: [1. a story that] [2. would explain] [3. my being here dressed like this] 1.subject 2. verb 3. complement. The main part of the complement (3) is the noun phrase 'my being'. As Meisam points out, you could replace 'my being' with 'my presence', which is more obviously a noun phrase. Could you also say 'me' here? Well, it depends how correct you want to be. That is, in fact, the word which a very large proportion of native speakers would use here. It's not grammatically correct, but it is very common indeed.

My being here kinda means my presence So if you consider it in that way, you couldn't say: "Me presence" so the same is true on "Me being here" "Being here" describes some thing that belongs to you

Interesting question! I learned from the answers. Thank you.

Both are NOT correct. My being is the correct form...