To publish a mission that has been saved as a draft, first go to the Mission Maker page. Scroll down until you see 'Drafts', and select the mission that is ready to publish. Then select the button in the top left, 'Edit mission'.
Once you're in the mission editor, check that all the mission details are correct on the final publishing page. Only when you're confident that all details are correct, click the big green 'Publish' button at the bottom right of the page.
Your mission is live!