To edit the details of an incident or to update its severity/status, you have two simple options.
Click on an incident title, and from the Incident Details Page, click on the pencil in the top right of the page. This will take you through to the Edit Incident form, from which you can change any parameters.
In the list of incidents, hover over any information that needs to be updated. Click on the blue pencil icon that appears to change the basic details, then simply click away to save.