If you are working in Appointment mode your calendar is already connected and busy time on your calendar is blocking your availability. Double booking is always prevented when working in appointment mode.
If you want to prevent double booking in Meeting mode make sure to connect your calendar and make sure that in the availability section of your Inbound settings the "Busy time blocks my availability" box is checked.
Please note that when an invitee selects a few times on your MeetMe page these times will not be blocked out since they have not been scheduled. Only when you schedule the time in Meeting mode will it become busy time on your calendar and block out your availability.