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I would suggest you first start with the Intel Edison Audio Setup Guide - download.intel.com/support/edison/sb/edisonaudio_332434001.pdf .
Check out this Adafruit forum post for reference but you'll also need a DAC (Digital-to-Analog Converter) codec chip in-between Edison and that Amplifier board. I2C isn't used for audio applications, but I2S is. Whichever codec you end up connecting up, you'll need to make modifications in the kernel to get things up and functional (see ASoC driver section in guide). When the system recognizes your codec, you'll be able to configure Alsa or Pulseaudio to use that codec as your 'sound card'. From what Adafruit offers, they have a 'combo' breakout that has both DAC+Amplifier over I2S with the MAX98357 chip (link). You should be able to find some Linux drivers available to get a head start.
You can reference Sergey's work on various posts on this site and also on his blog with the WM8731 codec - it's the most documented work I've seen thus far:
Others have gotten other codec chips working, feel free to search the forum and ask others directly for help once you have a codec you'd like to try and use.
I will build it this week. Thanks for all the detail.