To setup the hardware you'll want to make the following connections:
Don't worry about the exact pin connections to the Teensy. If necessary you can adjust in the code the analog input pin for the microphone, and the digital output pin for the neo pixels.
Finally, turn up the gain on the microphone amplifier to its maximum (about 1V peak to peak output) by turning the small trim potentiometer on the back all the way to the left.
This guide uses the Teensy 3.0 microcontroller for a couple reasons. Teensy 3.0 is a very powerful device that runs a full 32-bit ARM Cortex-M4 processor at 48 mhz. With such a powerful processor it's easy to sample audio and run an FFT in real time without resorting to low-level commands outside the Arduino/Teensyduino programming library. Furthermore the ARM Cortex-M4 core on the Teensy has native support for running Fourier transforms and other signal processing functions with the CMSIS DSP math library.
However you can still apply the principles and code from this guide to other microcontrollers like Arduino. Look for existing FFT libraries to give you the code you need for running a Fourier transform, and be aware of how quickly you can sample audio with the microcontroller. This tiny music visualizer guide is a great example of running an FFT and analyzing audio in real time on an Arduino.
Continue on to get the software necessary for this guide.