Why there isn't an "a" before the word "girl" in that sentence?

Well actually you are indeed correct there should be an "A" in the beginning of that sentence, if it were following an idea of some sort. In this case it is a description to the picture. There is only one girl, so "A girl" would refer to any girl, but "Girl" is being referenced in that picture since she's the only girl.

I agree with the above posters, and want to add that captions on photos are often fragments like that. It's a convention. It is probably to make the sentence shorter so it fits easily under a photo and can be quickly read.

Because its not a sentence, they're just showing you what is happening

To be grammatically correct you shouldn't start a sentence with a preposition.