On the "Race Details" screen, each race has a separate tab across the top. Click on a race and view the entries. Then simply click the round button in the row of the horse you're selecting. Once you have made your selections, click on "Save Picks" to save. You may enter all your picks at the same time for a single entry and clicking Save Picks at the end will save all selections. If you have multiple entries, you may select the All Entries tab which will display your entries. Again, select the race and horse for each entry and then click Save Picks.
For more information, view the tutorials under HELP on the homepage.