The most commonly heard version of "It's a Small World," the one played on the Disney ride, is actually a little hard to understand, I think, because it is sung by a chorus of childrens' voices. This version is sung by one of the composers himself, Richard Sherman. 

The song was intentionally written almost entirely with words of one syllable to make it easy to translate into other languages.

The song is a play on words with a triple meaning.

1) It is a wish for world understanding and peace: we are a small world, we are all neighbors, and we should treat each other that way.

2) The ride features the voices of children and animated, small, doll-like figures showing children of all nations; it's a world of children, a "small" world;

3) "It's a small world after all" is an English idiom or saying that is used when we are surprised by accidentally meeting people we know when we are far from home. You take a trip, and in the hotel lobby you see someone you know from home, and you say "It's a small world after all!" Or "Small world, isn't it?"

https://www.youtube.com/watch?v=iybDKRfAr-Q

It's a world of laughter, a world of tears
It's a world of hopes, a world of fear
There's so much that we share
And it's time we're aware
It's a small world after all

It's a small world after all
It's a small world after all
It's a small world after all
It's a small small world

The ride song, here with lyrics, goes on to a second verse:

https://www.youtube.com/watch?v=F9YqCP_B7EU

There is just one moon and one golden sun
And a smile means friendship to everyone
Though the mountains divide and the oceans are wide
It's a small world after all

It's a small world after all
It's a small world after all
It's a small world after all
It's a small small world