How does the approval process work? / How do rules get published in DTM?

DTM has a multi-step approval process. First, a change is funneled into an approvals queue where the details of the change can be viewed. The change can be viewed by clicking the 'Approvals' tab.

Once approved by an Approver, Approver & Publisher, or Admin level user, the change goes into the publishing queue where it remains until published.

To publish, an Admin, Publisher, or Approver & Publisher clicks 'Publish Property'. Once published the change is live. 

A change is always available in the Staging Library regardless of it is approved or published. A change must be approved and published to be available in the Production Library.

More information here: https://marketing.adobe.com/resources/help/en_US/dtm/publish_workflow.html