Why is the Chilean economy so much better than the rest of the countries in South America?
This is honestly a pretty complicated question - it's very difficult to pinpoint any exact factors that have led to Chile being so successful relative to its Latin American counterparts. I'm not particularly knowledgeable on the political factors, so I'll focus on some economic ones. 

As a developing country, Chile has several factors that have really helped with exports. 


Copper


Chile is by far the world's largest producer of copper. Copper is essential for electronics - meaning that Chinese factories have been buying massively - and Chile has been able to use its natural resources to boost its own economy. Copper is also far less volatile in price than oil; prices have only begun to go down because of slowing growth in China. 


Unique position in Southern hemisphere


Since the seasons are swapped between the northern and southern hemispheres, Chilean farms can produce fruits and vegetables to sell to the Americans in the middle of winter. If you live in the States, go to a local supermarket in the middle of winter and look at the fruits. Some of them (for me, I see grapes) are from Chile. 

Chile also exports fish. It has the longest coastline in the world, and has had some success in fish farming; both of these factors help Chile sell a lot of fish.

This is just part of why Chile has been successful (I haven't even mentioned government policy!) but hopefully it'll give you some background on why Chile has been more of a success story than other Latin American nations. 

Source: http://atlas.media.mit.edu/en/profile/country/chl/#Exports