how to save multiple checkboxes value in wordpress dynamically

Here you can just save the info in array and save that array in your post meta.

some thing like this will work;

$key = 'save-author-to-book';
$values_to_save = array();
$new_values = $_POST["save-author-to-book"];
$existing_values = get_post_meta( $post_id, $key, true ) ;

if(!empty($existing_values)){
   foreach($existing_values as $existing_value){
      $values_to_save[] = $existing_value;
   }
}


if(!empty($new_values)){
   foreach($new_values as $new_value ){
      $values_to_save[] = $new_value ;
   }
}

 update_post_meta( $post_id, $key, $values_to_save ); // you don't need to use if condition as  update post meta will take care

And here is how you can display the information stored in an array;

$key = 'save-author-to-book';
$values = get_post_meta( $post_id, $key, true );

foreach($values as $value){
   echo $value . '<br>';
}