こんな ↔ Such(or this much, that much)
いっぱい ↔ a lot of(or fully, bunch of)
黒歴史 ↔ shameful old memories (some of the past that you want to hide from others)
持ってる(=持っている) ↔ having(=have + ing)
~には ↔ to do ~ (or to be ~)
若すぎる ↔ too young
So I may interpret the sentence as the following.
He(or she) is too young to have such a lot of shameful old memories.
Remember that even though there is no 'he'(or she) which could have been the subject word in the original sentence, Japanese language is already implying it. The biggest characteristic of Japanese language is that they omit frequently with the subject(such as 私、僕、彼、あなた）word that depicts a person.
The meaning of 「には」can be totally changed depending on the context of the sentence. So my explanation can only go for the sentence you've posed on the question.