Why does Else smile at the end of "The Doll's House" by Katherine Mansfield?

The answer is that Else, who is the younger sister of Lil, felt happy because she was able to see the famous, miniature oil lamp that was part of the decoration of the Burnell sisters' doll's house. 

The Burnells, who were socially privileged girls, received the gift of a doll's house from a house guest, as an act of gratitude toward the family. The sisters were very excited about the doll's house, but it is one of the sisters, Kezia, who notices how unique and special this lamp really is:

But what Kezia liked more than anything... was the lamp. It stood in the middle of the dining-room table, an exquisite little amber lamp with a white globe. It was even filled all ready for lighting, though, of course, you couldn't light it. But there was something inside that looked like oil, and that moved when you shook it.

The Kelvey sisters, Lil and Else, were the exact opposite of the Burnells. The Kelveys were unkempt, neglected, poor, and disliked by their peers because of their lower status. Still, Else and Lil were equally amused and excited about the possibility of seeing the doll's house, but they do not get to see it. The Kelveys were not welcome anywhere and, as such, they only knew of the doll's house through the things that the other girls said. 

It isn't until Kezia invites the Kelvey girls over, against her mother's orders, that the poor sisters are finally able to see the doll's house, and the popular lamp as well. This is significant. It means that, for the first time, and for a brief moment before the Burnell's aunt, Beryl, kicks them out, the Kelveys were extended ONE opportunity that other, happier, and luckier girls also have: that of seeing the doll's house. It is also significant because Else and Kezia, two entirely different girls from different backgrounds, are still able to appreciate the beauty and uniqueness of the lamp with the same degree of interest. This is embodied in the phrase that Else says in the end: 

I seen the lamp

She, too, had an opportunity to see and appreciate something beautiful, even if only once.