Stop in for a cup of coffee

breaker vs GFI
Breakers (or fuses) are for a high current draw. They can open if the draw is between conductors or to ground
GFI's catch 'leaks'. They basically compare the power on both legs and will fail if not equal. The common reason for this is wet locations where a small current finds its way to a ground. They have a test switch that simulates this.
unfortunately neither can catch a bad connection. In this situation the amps are not any higher than normal so the breaker wont trip and unless the burn creates a path to ground the GFI wont see it.
Both can become less effective over time. If they are tripping when there is no problem they may be defective.

The burn in DNA's outlet looks like it may have started inside. The hole where the plug goes in is burnt. I would guess that the piece that engages the plug blade was loose. If you have plugs that dont hold tight in the socket this is what eventually can happen...