Why did Sesshomaru keep Rin? The answer is not so simple. Sesshomaru is a yokai hailing from the old era. Proud of his blood and power, Sesshomaru thought that humans were little more than a nuisance as they lacked anything remarkable about them.
To him, mankind was worth nothing more than a glance if they lacked the power to challenge him. Then the wheels started to shift once Rin appeared.
Why did Sesshomaru keep Rin? It's a simple reason.
Rin was only a child when she encountered Sesshomaru, but she did not fear him as he was injured at the time. Rin quickly gathered food from her village to feed him but he refused all help, as his pride showed as a yokai.
At one point, she splashed water on him, much to his surprise. It could be surmised that he found this intriguing, as humans don't typically go around throwing water at unholy demons.
Rin could not speak during this time due to a tragic event, yet her kindness showed through to him and anyone that encountered her. The girl was persistent despite Sesshomaru deep hatred of all things weak and fragile, yet his heart was starting to soften due to her unconditional attempts at kindness.
Later on she returns to her village, but is attacked by yokai wolves. Sesshomaru sees her corpse and is about to turn away until he remembers her smile. He resurrects her, this restoring her voice.
This is very important to explain why he keeps her in his company. The fact pertains that a human risked her life to nurse him back to health, even when she comes back injured from stealing food specifically for him.
This act of kindness unconsciously moved him to the point of no return. This could be counted as the first time a human ever smiled at him, not counting Inuyasha as he only holds a certain amount of mankind in his veins.
Rin wasn't the first pathway in Sesshomaru's path to accepting humanity as his father encouraged him to embrace the legacy of mankind. For years, he couldn't understand why his father would ever love a human or even try to live with one. It is not a old idea dredged up by an old man, but it's really a revolutionary idea for the time.
He saw a bit of his father's ideals in Rin and the idea is slowly moving into his very own. So Sesshomaru kept Rin because he started to care for her a bit and realized that Rin was an intriguing case and perhaps felt obligated to keep her since her persistence and kindness were too great for him to ignore.
Kindness to a demon is practically unheard of, yet she manages to break his icy demeanor little by little. Perhaps seeing a little girl struggle to nurse him back to help also aided in his perception of the human world. The absolute truth is that Rin is the start of his redemption arc of leading him into the art of humanity.