How can I hook this 8x8 rgb led up to an arduino?
You could use shift registers, however if you want to control the leds other than on/off, it is more convenient to have an IC with integrated PWM controller for all the LEDs.

For exapmle you could use TLC5947 or a similar IC. You can send serial information to it and the IC will PWM the LEDs to any intensity you like. It can also drive your leds with constant current, so no external transistors (or series resistors) are needed. 

If you don't like making a PCB for such an IC, Adafruit even sells breakoutboards for this particular IC.

You'll still need a shift register to control the rows however.