MySQL select datetime field where date equals today

You have to strip the time part of booked_at because 2015-08-05 09:10:56 is not equal to 2015-08-05. Try this:

select date(booked_at) from booking_dates where date(booked_at) = CURDATE();

To use index on column booked_at:

SELECT date(booked_at)
FROM booking_dates
WHERE booked_at >= CURDATE()
    AND booked_at < CURDATE() + INTERVAL 1 DAY