Perhaps the least understandable of all the fast days is Tzom Gedaliah. We fast in remembrance of Gedaliah who was killed during the times of the Churban Bayis Rishon. However, we do not fast for several other leaders or great people who were killed in cold blood. What do we fast for on Tzom Gedaliah?
After Nevuchadnetzar invaded Eretz Yisrael, conquered Yerushalayim, destroyed the first Beis Hamikdash, and exiled the vast majority of the people therein, Gedaliah was appointed governor of the few remaining Bnei Yisrael. Due to political disagreements, Gedaliah was assassinated by a group of Bnei Yisrael, and the remaining population fled to Egypt for fear of a fatal response by the Babylonian army. It is because of this murder that Chazal decreed a fast for generations. However, the ten tribes had already been exiled, the Beis Hamikdash had been destroyed, and the population had already been killed or exiled. Only a small group of Bnei Yisrael remained in Yerushalayim, and the man who was appointed to be merely a puppet leader, acting on the commands of his Babylonian superiors, was killed. Why declare a fast because of this – all had already been lost!
The importance of this event lies in the fact that the crime was committed the day after Rosh Hashanah. If a group of Bnei Yisrael were so unmoved by Rosh Hashanah that they murdered the next day, it is a tragedy that we must remember. The Radak holds that Gedaliah was murdered on Rosh Hashanah itself, which would make this crime even more heinous.
There is also a second approach to an answer:
The Gemara lists the events that occurred on the 15th of Av that caused it to be declared a day of simchah. One event was that we were finally given permission to bury those killed by the Romans in Beitar around the time of the destruction of the second Beis Hamikdash. When Bnei Yisrael arrived to bury the remains of the corpses, they saw that the dead bodies had not rotted nor decomposed but were complete and ready for burial.
What is the simchah in this event that appears to be a symbol of continuous destruction?
It was a joyous event because even when it seemed that Hashem had abandoned us, He still took care of us.
The events of Tzom Gedaliah were the opposite of this. We had lost the Beis Hamikdash and the vast majority of the population, but as long as there were even a handful of people left in Yerushalayim, we still had a vestige of the direct connection to Hashem that Yerushalayim offers. When Gedaliah was assassinated and the last handful of people fled to Egypt, we lost that last connection. That is why the event is so important and is the explanation given by the Mishnah Berurah (549:2): ‘The third of Tishrei, the day on which Gedaliah ben Achikam was killed…and the last flame (those who remained in the Land) was put out. For after Gedaliah’s murder they were all exiled, and many were killed.’
We can apply this idea to the Ten Days of Repentance. We must declare ourselves willing to follow Hashem’s path, no matter how big our commitment to improve is. Then we will retain our connection to Hashem.
The Shulchan Aruch rules that during the Ten Days of Repentance one should refrain from eating bread made by non-Jews. Why did the Shulchan Aruch highlight this halachah specifically; we anyway tend to keep this halachah all year round? The Shulchan Aruch teaches us that during the Ten Days of Repentance, it is of paramount importance to connect to Hashem and less important to make an impressive commitment to improve. For even a minor commitment to grow demonstrates that we are interested in connecting to Hashem’s path.
 Radak, Yirmiya 41:1. The Mishnah Berurah 549:2 seems to hold that Gedaliah was killed on the third of Tishrei.
 Gemarra Taanis 26b
 Shulchan Aruch Orach Chaim 603