DriveScore runs in the background and will automatically detect your trips. You don't need to do anything to get the app to start recording. Once a trip has finished, it will be uploaded and scored. The scored trip will appear back in the app in a few minutes.
If it classes your trip incorrectly (e.g., says 'Driver' when it should be 'Bus'), tap on the trip in the trip list to label it correctly. DriveScore works out your overall score based on the last 365 days of trips, so the mileage and score listed will be for the last year.