In order to enter scores, you will need to publish your schedule.
You will see score input boxes on each game starting the day it is scheduled (see screenshot).
Simply input the scores and hit Save Scores on the bottom right.
Standings will update automatically (refresh the page).
Once a match has a score, you can clear that score in its edit game dialog (click the little green icon above the team on the right).
Once a score is recorded, you can also change the points given to each team via the edit game dialog (for sports where points are a factor in standings, e.g. soccer, hockey).
The PRO/Tournament version allows you to designate other users as score keepers and enter results: see this article.