The relative clause that modifies きっかけ is 日本語に興味を持った excluding アニメが. And the subject of this 持った is not アニメ but 私 (which is implied; of course an anime doesn't have an interest in something). アニメ(が) is the subject of きっかけだ.
アニメが[(私が)日本語に興味を持った→]きっかけだ。
(literally) Anime is the trigger [←with which I got an interest in Japanese].
→ Anime is what got me interested in Japanese.
This is something you eventually need to get intuitively. Note that many English sentences would look equally ambiguous if you don't have the context and the knowledge of which word is used with which word. Please read my previous answer here, too. If you are feeling brave, take a look at questions tagged with ambiguous-relative-clauses.
In case you've missed it, you can read how this 普通に works here.
(~というの)もある is a way of giving multiple reasons. というの nominalizes the preceding clause, and もある says such a thing/situation exists. A very literal translation is "a fact/situation that ~ also exists".
- 時間がないというのもあるし、お金がないというのもある。
(It's not only because) I don't have time, but (also because) I don't have money, either.
- それも確かにありますが、一番の理由は彼女が努力したことです。
That's indeed one reason, but the main reason is that she made the effort.
アニメが日本語に興味を持ったきっかけというのもあったし、普通にアニメも好きだったんです。
(For one reason,) anime was what got me interested in Japanese, and (for another) I simply liked anime.