They need to publish their stories to an HTML file, and then submit that HTML file to you (plus any other files it may need, such as images). To do that, in the Twine window where you edit the story, you click the story name, then "Publish to File", and then save that HTML file somewhere.
The Twine story just exists in their browser's memory, so they'll need to do that from the same computer where they worked on their story. Because it's only stored locally like that, they'll need to publish it to a file so you can access it.
You can then import that HTML into Twine if you want to take a look at the source of a Twine story.
Hope that helps! :-)