This park is a glacier filled frozen island, and it's in the southern hemisphere...so it's winter there right now. What does ice do when it freezes? It expands. What does ice do in a crack when it freezes? It makes the crack bigger.
My guess would be that the freezing lake finally cracked through to another basin or cave, or that the rock along one side is fissured enough to allow water to run out of the basin faster than it runs in.
It's cool to see, because these kinds of things don't happen very often, but it's a great display of the impermanence of nature. Ice made the lake, and ice may have destroyed it.
On edit:
I found another article with pictures of the lake, which introduces another possibility:
http://www.usatoday.com/tech/world/2007-06-21-chile-lake-vanishes_N.htm. As you can see in the photos, the glacier extends right into the lake. If the glacier is receding, it may have exposed cracks or underground caves that had been previously plugged by the ice. Glacial lakes like this one also tend to form between the terminal moraine and the glacial head. If the glacier calved and caused a wave in the lake, it might have weakened the moraine enough to permit water through.