The format used to document the API is called API Blueprint – it's a standard for describing APIs. You can find the .apib files in docs/api/blueprint.
Those files can be imported in Postman or Paw for faster testing, for Paw this requires the API Blueprint extension.
It can also be imported into any service supporting the standard (Apiary, Dredd, etc.)
To run the tests via Dredd first you need to generate the documentation as seen above. Once this is done, all you need to do is run grunt dredd:
$ node_modules/.bin/grunt dredd