Specifically as a result of his mistake and pangs of remorse that surfaced in its wake, Reuven, much more so than his brothers, was sensitized to Yaakov's emotions and outright refused to do anything that would cause him anguish and grief.
The opening section of Parashat Vayeshev tells the unfortunate story of Yosef's sale as a slave to Egypt. We read that the brothers initially planned to kill Yosef as he approached them, but Reuven, in an attempt to save his younger brother, convinced them to instead kill him indirectly, by casting him into an empty pit. His plan was to later come and rescue Yosef. However, Yehuda then advised removing Yosef from the pit and selling him as a slave to passing merchants. The Torah tells that Reuven returned to the pit – to rescue him – and found that Yosef was missing. Evidently, Reuven was not with his brothers when they removed Yosef from the pit and sold him to the merchants. Rashi, citing from the Midrash (Bereishit Rabba, 84:19), explains that Reuven was "occupied with his sackcloth and fasting for having switched his father's bed." According to the Midrash, Reuven left after Yosef was thrown into the pit to focus his attention on his repentance for the sin recorded earlier, in Parashat Vayishlach (35:22), which Chazal understood as disrespectfully moving his father's bed into Leah's tent. (Reuven wished to defend his mother's honor, which was infringed upon when Yaakov moved his bed into the tent of Rachel's maidservant, Bilha.)
The obvious question arises, what connection is there between this incident, of the sale of Yosef, and Reuven's repentance for his sin?
One simple answer, perhaps, has to do with the meal the brothers conducted after throwing Yosef into the pit (37:25), and during which they noticed the merchants and decided to sell Yosef. When the brothers sat down for a meal, Reuven, who was still observing regular fasts to atone for his disrespect towards his father, left to continue his prayers. He was therefore not present when the brothers sold Yosef to the merchants.
The Beit Ha-levi suggests a different explanation. Even though Reuven succeeded in convincing his brothers to cast Yosef into a pit rather than actually kill him, and thereby hoped to rescue Yosef, there was nevertheless no guarantee that his plan would work. As Chazal famously comment, the pit was occupied by poisonous snakes and scorpions. The prospect of Yosef's death meant not only the tragic loss of a brother, but a loss of one of the twelve tribes of Israel. The Midrashim elaborate on the spiritual and symbolic significance of the number twelve as the number of the tribes; Reuven was thus saddened by the possibility that this number would now be reduced to eleven. This led him to feel that had he not interfered with his father's marital life, Yaakov perhaps would have begotten more children, in which case the number twelve could still be retained. Therefore, after the brothers cast Yosef into the pit in Dotan, Reuven returned to his process of repentance for his sin against his father.
A much simpler explanation was suggested by Rav Mordechai Gifter, in Pirkei Torah. Chazal emphasize Reuven's ongoing teshuva in this context because it directly impacted upon his response to the brothers' plan to kill Yosef. Reuven's preoccupation with correcting his wrong engendered within him a heightened sensitivity to his father's feelings. Specifically as a result of his mistake and pangs of remorse that surfaced in its wake, he, much more so than his brothers, was sensitized to Yaakov's emotions and outright refused to do anything that would cause him anguish and grief. It was therefore he who stood up for Yosef and attempted to rescue him. Having already insulted his father once before, Reuven understood the need to save his father from further grief, and herein lay the connection between his ongoing process of teshuva and the story of Yosef's sale.