〜てやる

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.