At the moment, Organizations are invite-only. This means you need to get in contact with a manager of that Organization in order to be added as a member.
To start, look up the Organization you want to join, either in the search bar or through Organization page. This will take you to the Organization's page.
For more information about finding Organizations, see this article.
Under the People section of the page, you will find a list of key members of the Organization. From here, you can look up one of those members in the Inbox interface, and manually make your request there.
For more information on sending messages, see this article.