"Exactly like when a charged object is discharged and it needs some time to charge up again"

That is exactly what you are hearing. The output is AC coupled though a capacitor to the headphones to block any DC component of the signal. With the headphones first unplugged the capacitor had one side open circuit so it stays discharged. 

When you plug in the headphones, the capacitor needs to charge to whatever the DC voltage is before it... and POP.. you hear that current as it energizes the speakers in the headphones.

Unplug and plug again, the capacitor retains that charge a long time.