The Home page is where you can get a glance at all Stories and Tasks assigned to you as well as all activities on the Stories, Tasks, and Epics that you follow. (More details on the Activity Feed here).
My Work consists of all Stories you own, while My Active Tasks will show Tasks within Stories that are assigned to you.
Story Due Dates and the Home Page
The Home page also surfaces a list of upcoming Story Due Dates and Stories that were due in the previous month, which can be filtered either by all Due Dates throughout your Shortcut or on Stories that are owned by you.
Editing Stories from the Home Page
Clicking on Story cards, titles, or numbers will open them in a dialog where you can make edits as needed, and clicking on the checkbox on Tasks will mark them as completed in the Story.
You can also edit multiple Stories from the Dashboard by clicking the checkboxes in the upper-right hand corner of the Story and then selecting the Edit button that appears above.
The Tasks that appear on the Dashboard are only from Stories that have not yet been completed.