Understandig 4x4x4 led cube code

Hello,

I have built myself a 4x4x4 led cube and I want to control it using my Arduino Uno.
I have seen a lot of people not using transistors while they still light up the entire cube.
This guy says that you can ditch the transistors because you will only be lighting up one layer at a time,
but at about 10mA per LED you would still blow the c*** out of you microcontroller!
I wonder if it has something to do with this code everyone is using:

https://dl.dropboxusercontent.com/u/21892130/121118%20-%20LED%20Cube/LED_cube_4x4x4/LED_cube_4x4x4.ino

Could it be that this lets you light up only one LED at a time and multiplex that very fast?
If not, how can everybody get away by not using transistors and still have a nice bright fully lit cube?

Best regards,

Polo9n