What is Yehudah telling Yosef here?
Ramban and Seforno: Yehudah was telling him that he preferred to become a slave permanently, than to return to his father without Binyamin and witness his suffering - as he was bound to weep and mourn his loss incessantly.
Ohr ha'Chayim #1: Even without the reason that I will sin to my father forever, I cannot return without Binyamin due to the pain it will cause to my father.
Ohr ha'Chayim #2: This explains why Yehudah said (44:16) that all of them will be slaves. If they will not ascend to their father, they will not see the evil. 1
Ohr ha'Chayim #3: If you will say that if I do not return Binyamin, Yaakov will not be cruel [to let Yehudah be in Niduy], and he will permit the Niduy 2 - still, I cannot return due to the danger to my father.
If none of the brothers return within a reasonable amount of time, Yaakov will be even more anguished. He will think that they were killed for being spies! (PF)
Ohr ha'Chayim: This is like the claim of the Ishah Chachamah who told David that one son killed his brother - if the murderer is killed, this only increases her pain (Shmuel II 14:7)!
Why did Yosef need to hear this?
Ramban: Otherwise, he might have suspected Yehudah of volunteering to take Binyamin's place, because he was more adept at escaping.
Why does Yehudah discuss seeing the evil that will befall his father?
Ha'amek Davar, Ma'aseh Hashem: Also if I stay in place of Binyamin, perhaps my father will be unable to bear my loss, and will be concerned lest I die due to his sin - but I will be spared from seeing his pain. 1