How does this blinking LED circuit work?
There are two possibilities:

The LED is really an integrated unit that contains the flashing logic.  Such things exist.  You apply power, and it flashes.

There is an IC under the white blob.  This is a high volume but low cost production technique called chip on board or "COB".



I think #2 is more likely, especially considering the mass-produced high-volume look of this device.  LEDs with integrated flashers are relatively rare and therefore expensive.  A dumb flashing chip, especially without package, can probably be had for a few pennies in high volume.