Click on any vessel to open her Vessel InfoWindow and, then, click on the Past Track button.
You will be able to view the subject vessel's past tracks on the Live Map. The tracks will have a matching colouring to the vessel' icon for easier identification - blue for passengers' vessels, red for tankers and so on.
The displayed tracks are derived from each vessel's position record. Notice the arrows along the track-line. Hover your mouse's index over any such arrow and you will be informed regarding the speed, the course and the timestamp of the respective position/record.
You can zoom out to view the entire vessel's track on the Live Map or zoom into the positions that you are more interested in.