The following is just a theory I have, and if you think it should be moved to the comments, I'll do that.
Based on the understanding that Tarsia is Tarshish or Tarsus, we can understand a bit of the hatred Bigtan and Teresh held for the king.
According to Wikipedia, some modern opinions are that Achashverosh was King Artaxerxes II, who at one point, waged war against his brother, Cyrus the Younger. Tarsus was the capital city of a region call Cilicia, eventually a satrapy (province) (from satrap, or achashdarpan (אחשדרפן) in the Tanach) in the Persian-Median Empire. Cilicia chose to side with Cyrus the Younger, who eventually lost to Artaxerxes II (an event that's possibly referenced by Rashi on Esther 1:2: "When King Achashveirosh had established himself, etc. When the kingdom was firmly established under his control."). Prior to the war, Cilicia had a vassal king. After siding with Cyrus the Younger and losing, the vassal king was dethroned and Cilicia was turned into an ordinary satrapy (province).
Having been internationally disgraced by Artaxerxes/Achashaverosh, it's no wonder that Tarsians such as Bigtan and Teresh were out to get him. The emphasis on them being both Tarsians and speaking the Tarsian tongue may be there to tell us that these were Tarsian nationalists; zealous extremists plotting to assassinate a powerful enemy monarch to avenge the honor of their country.