Most of the secret sauce is in the firmware. You can find it here: https://github.com/LarcomJ/audioSpectrumAnalyzer on github.
The easiest method to recreate this would be using a psoc 5 development board "cy8ckit-059". Unfortunately, I believe these are "end of life" products and I will be porting the code over to the stm32 micros. Please excuse the sloppy coding, dont code like me =D
The code is written to drive a led strip utilizing the APA102C which relies on the SPI interface. If you install the psoc creator and open the project there is a schematic view that should help explain the port connections to the led strip and the microphone.
I plan to add more info as i have time. If you have any questions feel free to ping me.