Oak is a command-line application which takes a proof file as input, and tells you whether or not the proof is correct.
Oak requires the Ruby programming language. If you don't have Ruby, go here to find out how to install it.
Once you have Ruby, to install Oak, just run:
gem install oakproof
To test that the install worked, run
oak -vwhich should print the version number of Oak.
Once Oak is installed, you can follow the tutorial.
You may also want to download the examples.
The Pulsar text editor (formerly Atom) has a package language-oak which provides syntax highlighting, automatic indentation, and comment toggling for Oak. (Similar packages for other editors would be welcome.)
If you run into a problem, you can contact Tim Smith or post an issue on Oak's page at GitHub.