Elsa always loved Anna. She was her little sister and she cared for her even as a child. When she was eight years old, she always had time for sister, even at night. She liked to play with her powers to amuse Anna. When the tragic accident happened, Elsa was creating snow playground for her and was trying to make her laugh. When she hit Anna with her powers, she was devastated. What a horrible news was for her that Anna's memories were erased and she couldn't play with her anymore. This was adding to her loneliness a new level.
While growing up, Elsa always feared she may again hurt her sister. She never let her come closer to her, because that was the only way she knew to protect her. She was protecting little sister even as an adult. While chatting with her on coronation ball, she let herself at loose at the moment but she reminded herself that she has these destructive powers and stepped back from her.
It was always fear before hurting Anna that drove Elsa's distress. She ran away, because she was sure that her powers may hurt her. When confronted with sister in ice castle, she was at first happy that she sees Anna but the fear was still there. She would love to be normal sister but the fear was stronger. She pushed her away because she was unsure about her powers.
When Hans told Elsa that she killed Anna, she was almost ready to be killed by him, that devastating were the news. She always blamed herself for hurting Anna in the past and the knowledge that she hurted her once again, this time to death, was too much for her.
This was true sister love that brought Anna back to life. This love was stronger love than Kristoff's and Anna's love to each other. Nothing could compare to true love of two sisters who lived separated for too long and needed one kick to rebuild whole lost time. The movie ends with Anna and Elsa playing together at ice - the powers of Elsa aren't destructive anymore and she can love her sister without bad consequences.
After losing her parents and knowing that at the same time Anna lost them too, Elsa wanted to help her, hug her, protect her - yet she stayed in her cold room, making her sister believe that she doesn't like her anymore and what was in the past, was only the past. Yet in the end, she made more to Anna. She showed her - truly - how strong her love for her is and how much she can do for her now - if allowed. In small additional movies created after Frozen, Elsa always was trying to cheer up Anna or make her happy. That was her redemption and her gift for the most beloved person she ever had.