What "begin to flake" means in the context of sauteing a fish?
When the fish cooks, the meat will start turning opaque; at one point, you will see that the fish will start to "split" into flakes or layers.

see this video (at about 2:15 and onward) you start to see layers in the fish.

https://www.youtube.com/watch?v=gluRaUv88sk

For example in the following video, you see the salmon has "layers" of fat and meat; when the meat is just right, the layers will be easily "flaked" when the fish is cooked.

https://www.youtube.com/watch?v=3KLQnGS1TjQ