How does the led fade circuit with the NE555 timer work?

You're fairly close with that analysis.

Q1, the transistor is used as an emitter follower, in this configuration it only accepts a small current into the base. this is because the transistor conducts so that the voltage on the emitter rises until it's close enough to the base voltage to prevent most of the current from flowing into the base.

C1 is charged and discharged mostly through R1

The varying voltage on the capacitor causes a varying voltage at the emitter which causes a varying current through R2 and the LED.