People should probably check the pictures of the water on your cited webpage before they take this as some kind of evidence. The lakes listed seem like mostly geothermally heated, or no longer existing.
Yes, and that page says "There are many catchment areas high in the mountains that are capable of collecting liquid water, and some of these are now filled with snow and ice. As the climate gets warmer, it's probable that they will melt out and a new high lake will emerge."
Hence, new high lakes may be evidence of warming, which might be from climate change, or from volcanic heating, or changes in weather patterns causing more rain.
Some examples of new lakes:
"Imja Lake is forming by glacial melting and did not even exist in 1960."
"Ridonglabo Lake" is a classical moraine lake produced by climate warming. The glacier retreated from its terminal moraine, leaving a depression in which glacial meltwater could collect into a lake. We know this happened between 1925 and 1988 because we have two maps of the area from those years.
"Laguna Glaciar lake is a glacial lake near Sorata in Bolivia at 5,038 meters above sea level. The lake has gotten much larger during the last 50 years due to the warming climate in that area."
You wrote "How does this prove anything?"
Assuming you aren't asking the abstract question "What is proof?", it contributes support to the widely-held understanding that global warming is occurring.
Limited support, IMHO, when taken as a standalone data point, as local climate does not always reflect global climate.
Because this shouldn’t be the case so high in the Alps. Snow melts on my roof every year, it does not follow that the polar icecaps should also melt.