What software is needed to run this file extension?
You will need GNU Guile and Guile-SDL2 installed on your machine and then follow the install instruction laid out in the README. Most Linux distributions should have both packaged; if not, you can install GNU Guix to get them.
Assuming by "this file extension" you mean ".tar.gz", you're in for a wild ride. Under Linux, you'd just need "tar" and "gzip", which come out of the box with pretty much any distro. Under Windows, there are some alternatives to those, but you will need MinGW or Cygwin to build and install the software anyway. Since MinGW does not package Guile 3.0 at the moment (as far as I'm aware), it would seem, that Cygwin is your only option.
TL;DR:
- Windows only: Install Cygwin [or MinGW once it does package Guile 3.0]
- Install GNU Guile and Guile-SDL2
- Unpack the source tarball
- Follow the install instructions in the README