the storyline in the anime moved ahead of the manga it was based on
「原作」= the original / the original manga / the manga it was based on
「追いついちゃう」= overtake / move ahead of ~
「アニメが原作(漫画)の話に追いついちゃう」= the anime story overtook the original manga / the storyline in the anime moved ahead of the manga it was based on
「アニメ」と「漫画」はもう英語になっていますのでそのまま使ってもオッケーです。