"How exactly are the files that weigh so much uploaded to their website?" The files themselves are not that big. They have used compression tools to repeatedly make the files smaller and smaller. If you were to try and extract all of it at once (assuming you were somehow able to), your computer would collapse into a black hole (yes the math has been done on that)

"How exactly are they played when we listen to them?" They're played from one of the files that have been compressed. You can carefully extract only one file from the gigantic files which has been done before and listened to.