At Thorin's words, Kili felt some tension he didn't know he was holding start to melt away. The fact that Thorin was happy here made Kili happy and made him hopeful that their family could continue to make a life in this amazing place. He'd be happier if more of their people were here, yes, but he was very happy being here with Fili, Thorin, and Bilbo.
"Uncle, I very much want to make a life for myself here. There are so many things to experience and learn here. People are happy here and... you're smiling again. I didn't realize how much I had missed seeing you smile until I saw you do it again."