Because they were rivals
The fight had come to an end. Unexpectedly, Renji's struck had been caught by Kurosaki Ichigo with his bare hand. Ichigo's struck was just too powerful for the red-haired shinigami to catch. Ichigo was just too powerful.
He had thought that with training he would be able to beat Abarai Renji easily. But Renji hadn't been slacking off either. Besides, his powers were supressed when in the human world, so he was much stronger than before. And Renji was already strong.
Renji remembered when he first met Rukia. A little girl in the Soul Society whom he had asked to join his group. Even though she was the girl, she was also the tough one. Rukia would always be the one to get Renji out of trouble. In the end, Rukia had lost her powers and was condemned to death and Renji had made to vice-captain.
Ichigo remembered when he first met Rukia. A shinigami looking for a hollow that had come to the human world with the intention to eat the orange-haired boy's soul and kill away all his family. In the end, it was Ichigo that purified the monster's soul and Rukia only stood there, powerless, watching the battle.
Renji had always thought of his relationship with Rukia. At first, when he met her, the boy was enfatuated with the daring little girl. He could tell that she was the one. But as time went by and their friends died, the feeling faded away. She was just Rukia, his friend and comrade. When he was promoted to vice-captain and got the news of Rukia's absence, they were just former shinigami and present vice-captain.
Ichigo never really thought much of his relationship with Rukia. At first, they were former shinigami and substitute shinigami. Then, a friendship was born and Ichigo and Rukia would always walk together, just each one's presence being enough. In the end, Ichigo started getting different vibes, different feelings awoke inside him and the mere thought of Rukia was enough for him to be enfatuated.
Renji was the one who managed to delay Rukia's execution. He feels that Rukia owes him. But he doesn't want her to pay.
Ichigo was the one who caused Rukia to be condemned to death. He feels that he owes Rukia and he wants to pay her.
Though Renji followed his captain's orders and attacked Ichigo, he knew that Ichigo was the one who was right and not Kuchiki Byakuya. He wanted to save her, but now he couldn't.
Though Ichigo was ordered by Rukia not to come and get her, he knew that he himself was the one who was right and not Kuchiki Rukia. He wanted to save her, and he would.
"Please! You have to save Rukia!"
"Yeah..."
