By - alexadb123
The bulb has a tiny power supply unit built into the plastic stem part. It can be pretty simplistic (just a capacitive voltage dropper) or something matching or exceeding the complexity of a phone charger (switch mode PSU with dimming functions, maybe even Bluetooth or WiFi radios and internal memory). The more complicated ones have microcontrollers inside that boot up and run a program which takes some time before the controller is ready and activates the LEDs.
This doesn't explain shit. You just said it has capacitors and that when you turn off the light, the capacitors store electricity to run the LEDs for a short time before discharging. That has nothing to do with the LED having a delay when you turn it on. You're implying a delay when you turn it off, but that doesn't happen on any LED I've seen.