Likely the toy has a 4060 chip (or similar), when  you turn on the switch it is connected to the 2 terminals that give it the slow flash, hit the switch again and it sends power to the next set of terminals on the chip to increase the speed of the flash, hit the switch again, so on, and so on.

With a background in electronics we used to play with these in the lab and with the right LED's and flash speed,  you can create quite the hypnotic, psychedelic effect.

Unfortunately I have forgotten quite a bit of it, none of which would take much to refresh my memories, but got to like math and physics.