The general hardware setup is as the following picture. You do need a proper programming header connected for downloading, or use NodeMCU board. In this project I'm using my own developed ESP8266 breakout board, and everything is setup on a perfboard.
I have put all source codes on https://github.com/baoshi/ESP-Audio-Spectrum
The ESP8266 starts as AP. Connect your computer to ESP-xxx WiFi Access Point and type 192.168.4.1 in the browser to see the demo. Check the linked video below for demonstration.