Why is the Chilean economy so much better than the rest of the countries in South America? Why does Chile have a better economy than all of the other countries in latin America? 


It is labeled as a high income country by the IMF, 
It is the only Latin American country that is part of the US Visa Waiver Program
It is one of two Latin American countries that is an OECD member.
Unlike some of the other developed countries in South America,Argentina , they don't have chronic inflation problems.


What was is that set Chile apart? I've read a little bit about how "The Chicago Boys" played a big part of the country's success. What did they do differently in Chile? Didn't a lot of the policies that they helped to put in place take place under Pinochet? I read that Chile's chronic inflation problem was also solved under then Pinochet regime. If so, how do you think that affects the legacy of Pinochet?
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