Quote 1
"Oh, children," said the Lion, "I feel my strength coming back to me. Oh, children, catch me if you can!" He stood for a second, his eyes very bright, his limbs quivering, lashing himself with his tail. Then he made a leap high over their heads and landed on the other side of the Table. Laughing, though she didn't know why, Lucy scrambled over it to reach him. Aslan leaped again. A mad chase began. (15.40)
Part of Aslan's nature is playfulness. After he is revived by the Emperor's magic, Aslan romps with Susan and Lucy, and all three simply enjoy the feeling of being alive and playing together. There is something simple and beautifully good about their play.
Quote 2
"But where is the fourth?" asked Aslan.
"He has tried to betray them and joined the White Witch, O Aslan," said Mr. Beaver. (12.17-18)
It's interesting that this conversation takes place. Aslan pretty much knows everything that is going on in Narnia, and he definitely knows what's up with Edmund and the Witch, so we assume that asking about the fourth child is just a formality. It needs to be said outright that Edmund is a traitor.
Quote 3
"Hand it to me and kneel, Son of Adam," said Aslan. And when Peter had done so he struck him with the flat of the blade and said, "Rise up, Sir Peter Fenris-Bane." (12.27)
Aslan knights Peter in order to recognize his transition from boy to warrior. The actual knighting ceremony doesn't cause the transformation – it just makes it official.
Quote 4
"It means," said Aslan, "that though the Witch knew the Deep Magic, there is a magic deeper still which she did not know. Her knowledge goes back only to the dawn of Time. But if she could have looked a little further back, into the stillness and the darkness before Time dawned, she would have read there a different incantation. She would have known that when a willing victim who had committed no treachery was killed in a traitor's stead, the Table would crack and Death itself would start working backwards." (15.38)
OK, we admit that this sounds like a great big "Ah-HA, gotcha!" to the Witch. Aslan knew a loophole that nobody else did, and it fixed everything! But the spiritual point is that one great sacrifice can atone for someone else's treachery. One victim can stand in for another, and, by doing so, he can free the whole world.
Quote 5
"Here is your brother," he said, "and – there is no need to talk to him about what is past."
Edmund shook hands with each of the others and said to each of them in turn, "I'm sorry," and everyone said "That's all right." And then everyone wanted very hard to say something which would make it quite clear that they were all friends with him again – something ordinary and natural – and of course no one could think of anything in the world to say. (13.27)
If Edmund were our brother, we're not sure we'd be able to forgive him so quickly and easily. It seems like Peter, Susan, and Lucy are just so glad that the Witch didn't murder him that they've forgotten he put them in mortal danger only yesterday.