The timer divides the Dark Hours raid into five stages, one for the initial approach on the tarmac, and then the four raid bosses. When you launch the raid, or are about to start any of the stages, click the button above the timer, which will be labeled for what the next step currently is. When you complete a stage, click the same button.
If you wipe during a stage, you can click the "Wiped" button to keep count of how many times the team wiped for a stage.
The timer is very basic at present, so there's no way to edit the times if you make a mistake. If you need to reset the timer, just reload the page. No data is retained or saved.