Issue with data array format

Your code looks good though. But Try the following approach –

$current_date = $_POST['current_date'];
$current_date = date( 'Y-m-d', strtotime($current_date) );

$post_id = (int) $_POST['post_id'];

$query = $wpdb->prepare( "SELECT start_date FROM {$wpdb->prefix}booking_history WHERE start_date=%s AND post_id=%d", $current_date, $post_id );
/* get_col used to retrieve column records */
$results = $wpdb->get_col( $query );

echo '<pre>'; print_r( $results ); echo '</pre>';