Hello.
1) If you need to auto-approve each booking, so then its possible in the Booking Calendar Business Small or higher versions to activate this option at the Booking > Settings General page in Auto cancellation/approving section.
2) If you just need to receive new bookings, without needs to block days in calendar at all, so then in any versions of Booking Calendar (including Free) you can activate this option “Allow unlimited bookings per same day(s)” at the Booking > Settings General page in Advanced section. Your visitors will be able to make unlimited bookings per same date(s) in calendar and do not see any booked date(s) of other visitors.
3) In general even, if you will not use approving of booking, for visitors at the front-end side its only matter of how is showing such days (I mean approved and pending – different background color). So if you would like to get bookings and set such days unavailable, without approving you can do this. Its means that you will have in calendar “pending days”, and these days will be unavailable. And in calendar legend section, you can deactivate showing “approved” element item (under calendar), and rename title of “pending” legend item. You can rename it at the Booking > Settings General page.
Kind Regards.
