Here's some info a sent to someone before (I really need to add this to the docs):
To explore the API you can append /api/ to most URLs, for example:
To get results for a query you can use the following:
Other resources have API access too, but I assume you need access to the queries. If you have something else in mind, let me know.
If you try the above in the browser, assuming you already logged in to the demo instance, it will just work.
To get it working from a script, you need to add an API key to the request (example:
http://demo.redash.io/api/queries/4?api_key=60f0c7364ce547a49b0a297aa549c29bcfcf9db2). Each query has its own API key (which you can see if you click on the key icon in the query page), which lets you access this query and its results using the api. Also each user has an api key, which gives you access to all API resources (you can get yours by browsing to /users/me#apiKey).
API keys per user available from v0.6.3. And the UI to show them available from version 0.8.
Before v0.7 you had to sign requests rather than just add api key. You can enable api keys instead of signatures in older versions too, let me know if you need details.