1. If Edison is OFF and +5V USB is plugged in. That will turn ON Q1.1 and turn OFF Q2.1 resulting in logic high for PWRBTN# on Edison. Basically that will do nothing for Edison and it will stay OFF.
2. Note that since the Edison module is OFF, the gate of p-channel FET Q1.2 is 0 volts and the pin 2 of R4 is at VBAT while pin 1 of R4 is ground because Q1.1 is ON.
3. Now removing the +5V USB will make the gate of Q2.1 to VBAT thereby turning it ON. This will pull down PWRBTN# pin on Edison module.
4. Once Edison is turned on, net V3P3_Edison will become 3.3V turning OFF p-channel FET Q1.2. This will lead to the gate of Q2.1 pull down to GND by R5. This will bring back the PWRBTN# of Edison high.
5. Note that once Edison is turned on, if you insert +5V USB, nothing will happen.