To Humiliate
To Embarrass
Because the hiragana ends with an う sound, you know this word is a verb. The kanji itself means humiliate so the verb vocab version is to humiliate.
はずかしめる
Remember the word 恥ずかしい? It has the same reading as this one, and a similar meaning as well.
みんなの前で私の事を辱めるのが、どうしてそんなに面白いんですか？
Why do you find it so amusing to humiliate me in front of everyone?