The app requires you to fully complete the current day of a multi-day program before you can move on to the next session. In the Meditate menu, tap on the name of the program and you will see a circle for each session on the program page. You will know that a session has been completed when the circle for the day is filled in with green.
If you do not see a green circle for the session, this means the session did not register as complete. To move on to the next day, simply let the session play through one more time. Once you see that the circle is green, you should be able to tap the circle for the next day and listen to the session.