The Midrash (Esther Rabbah 7:8) teaches that, by answering that he is a Jew, Mordechai really intended to emphasize that, as a Jew, he is forbidden to worship anyone or anything besides H-Shem.
Rav Shlomo Kluger says that “Mordechai’s words” indicate his reporting the plot of Bigsan and Seresh. Mordechai wanted to see if his demonstrated loyalty to the king would be enough to excuse him (and perhaps the other Jews) from this bowing.
The Chasam Sofer says that the words “that he is a Yehudi” refers to Haman. As mentioned in the Talmud (Megillah 15a), Haman sold himself as a slave to Mordechai. Yalkut Shimoni (953) tells us there was rebellion against Achashverosh in one of his Indian states. Haman and Mordechai were chosen to command two of Achashverosh’s battalions. Due to his spending practices, Haman ran out of provisions. Mordechai, due to his righteous care for his resources (see Rashi to Bireishis 32:25 and Talmud, Chullin 91a), did not. Haman begged Mordechai for some of his rations, on condition that Mordechai sell himself to him as a slave, to which Haman agreed. Having nothing on which to write handy, Mordechai wrote the deed on his shoe, or armor he had on his feet. That being the case, a slave to a Jew who then goes free becomes Jewish, himself (Talmud, Chagigah 4a and brought down in Halachah in Shulchan Aruch, Yoreh Deah, 267:3-5, 11). According to the Chasam Sofer, then, Mordechai was saying that he does not have to bow down to him since Haman was once his slave. For that reason, according to the Midrash, every time Haman would pass by, Mordechai would point down to his shoe.
The verse makes it sound as though the servants did not trust Mordechai, and Mi’archei Lev writes that Mordechai gave them reason to respond this way. After all, it was well-known that he was from Benyamin, but he aroused suspicion by saying he was a Yehudi.
Rabbi Yehonasan Eibshutz writes that Haman felt confident about conquering Mordechai as he was from Benyamin. Here, Mordechai is pointing out that he comes from another tribe as well – Yehudah. Yehudah, being the tribe of Moshiach, is the great challenge to the power of Amalek. Mordechai represents the Yehudi who can conquer the power of evil. Rav Eibshutz also writes that Haman set up a test for Mordechai by one time coming out without a statue. Nevertheless, Mordechai still refused to bow to him. Even though Mordechai knew there was no statue, other people didn’t know, and this would constitute maaris ayin.