話せなかった自分が悔しくて勉強を始めたって英語でなんて言うの？

When I couldn't speak English in front of everyone, I felt so frustrated. So, I started to study.

まず最初に背景を言って、→( When I couldn't speak English in front of everyone), みんなの前で英語がはなせなかった時、I felt so frustrated すごく悔しかった。So I, I started to study （だから、勉強をし始めた）

So, I started to study を　So, I decided to studyでもOKだと思います　（勉強することを決意した）。

I hope this helps!

~~始めた - started
Started studying
ちょっとスラングっぽいけれど”kicked off”もあります。
Kicked off my studies - 勉強を始めた

話せなかった - could not speak, wasn’t able to speak

悔しい - feel sad, feel upset
自分に悔しい時は - upset at myself を言える
“I was so upset at myself”

くて - この場合は”because”とか”therefore”(なので)の意味です
I was upset so I started studying
I started studying because I was upset