〜てやる

Meaning

The base meaning is doing something for an inferior or junior party, or for animals and plants, etc. (e.g., 弟に買ってやる、花に水をやる). From this comes a usage expressing the speaker's strong will, especially a determination charged with defiance, retaliation, or competitive spirit. In this volitional usage, with the feeling of 「今に見ていろ」, it is used as 「見返してやる」「辞めてやる」. The latter is casual spoken language.

Examples

弟に自転車を買ってやった。 I bought my younger brother a bicycle.
いつか必ず見返してやる。 Someday I'll definitely prove them wrong.
こんな会社、辞めてやる。 A company like this—I'll just quit.
絶対に成功してやると心に決めた。 I made up my mind that I would succeed no matter what.