This quote is supposed to be about Brunhilde singing her final aria at the end of Gotterdammerung, which is the last opera in the Ring Cycle (four total operas) and is about 6 hours long. (!!!) Brunhilde is often a larger woman, and her traditional costume has her in a chestplate and a winged hat - she's actually the source of that hoary old image of opera singers.

The phrase is very modern, a lot of it has to do with Wagner's domination of the opera canon in the 20th century, as well as his domination of what people think opera is in American popular culture. Baroque operas and early classical operas typically ended with a chorus, not an aria, so the phrase couldn't come from that time. Some late classical/Romantic operas have soprano aria finishes, but it's not really common enough to get you the phrase "it ain't over till the fat lady sings." Fair amount of opera singers have always been fat though, I'll give you that one! :) But they just don't always finish operas, not even close.

Wagner's operas inspire some funny quotes though. Here's a few:

"The opera started at 8pm. At midnight I looked down at my watch. It said 8:15." -- attributed to Billy Wilder, but dubious.

"After the last notes of Gotterdammerung I felt as though I had been let out of prison." -- alleged to Tchaikovsky

"Monsieur Wagner has good moments, but awful quarters of an hour!" -- Rossini

"In the evening we went to see Tristan und Isolde. That was the most disgusting thing I have ever seen or heard in my life. To be forced to see and hear such crazy lovemaking the whole evening, in which every feeling of decency is violated and by which not just the public but even musicians seem to be enchanted - that is the saddest thing I have experienced in my entire artistic life. I remained to the end since I wanted to hear the whole thing. During the entire second act the two of them sleep and sing; through the entire last act -- for fully forty minutes -- Tristan dies. They call that dramatic!!!" -- Clara Schumann (this one is not dubious at all!)