What happened to the limewater in
the experiment? What does this
prove?

Explanation:

In the reaction above, we clearly see that a chemical reaction took place but what kind of reaction you might ask?

Looks like it is Thermal decomposition of copper carbonate because decomposition reaction takes place due to the added heat. Decomposition in simple terms in chemical breakdown and here this result is obtained by adding heat.

1) What happened to the lime water?

Although not pictured above, but my assumption is that lime water turned milky or turbid because when CO2 comes in presence of limewater they react to form a percipitate of Calcium carbonate which is the milky color that you get.

2) What does this prove?

- My understanding would be that it proves that CO2 was formed, and that most metal carbonates undergo thermal decomposition into metal oxide and carbon dioxide, and also that a reaction took place since new products were made.