I need an efficient way to write data from a select (multiple) field form to MySQL PHP

Can you be a bit more specific?

What does your database structure look like?
What does your form data look like?

The more information you can provide the easier it will be to understand your request.

As a practical matter, software efficiency is irrelevant  when you're talking about getting external input from a client form.  Clients respond in multiple seconds.  Disks rotate in a few milliseconds.  Processors run instructions in some small number of nanoseconds.  So your focus should be on the user-experience, because it's about three to nine orders of magnitude more time-important than what happens after the data reaches your server!

The best way is to write a store procedure. IF you can give me more details I can write it for you !!

I am assuming you mean something like this

  <form action="reflect.php" method="post">
    <select multiple name="data[]">
      <option value="1">One</option>
      <option value="2">Two</option>
      <option value="3">Three</option>
      <option value="4">Four</option>
      <option value="5">Five</option>
    </select>
    <input type="submit" />
  </form>

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Note the name of the field in the above example is data[].

To explain how POST/GET data is presented in the case of a multiple select
In a POST with a select with attribute multiple values are submited as (assume name="controlname")

controlname=value1&controlname=value2&controlname=value3...

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When PHP is asked for $_POST['controlname'] - it will return the last value with that id - in this case value3.
To get access to all the data specify the name with [] at the end - this will result in a POST that looks like this

controlname[]=value1&controlname[]=value2&controlname[]=value3...

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Now when you ask PHP for $_POST['controlname'] you get a variable back that is an array with all the values that had controlname[] as the key. You can now iterate through that array using a foreach or similar.

$data = isset($_POST['data']) ? $_POST['data'] : false;
if ($data && is_array($data)) {
   foreach($data as $item) {
        // append to Query here
   }
}

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In an online forum like E-E It's often difficult to discern the level of expertise that underlies a question.  If you need general background knowledge, you might want to start here.  If you already have some familiarity with HTTP requests using PHP and MySQL, and you're looking for a table maintenance script, this article might give you some good ideas:
https://www.experts-exchange.com/articles/12335/PHP-and-MySQLi-Table-Maintenance.html

Based on the comments above this is what I came up with so far:

$tags = isset($_POST['tags']) ? $_POST['tags'] : false;
        global $database;
        $pre_sql = "delete from contact_tag where contact_id='$id'";
        $result_set = $database->query($pre_sql);
        if ($tags && is_array($tags)) {
            foreach($tags as $tag) {
                $id = !empty($_GET['id']) ? (int)$_GET['id'] : $newContact->id;
                $sql = "INSERT into contact_tag (contact_id, tag_id) VALUES ('$id','$tag')";
                $result_set = $database->query($sql);
            }
        }

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Is this the best way? Also is there a way to make this code a bit more reusable?

You mean putting a query before the delete to verify if they correct?

Also in terms of the delete is there a better way to check if they exist and entering or deleting the values that correspond to what the user input?

better way to check if they exist and entering or deleting the values that correspond to what the user input

Can you elaborate.
What is the objective?
When they submit must all previous combinations for this contact_id be removed and replaced with the new ones?
Or are the new ones meant to add to what is there?

I am assuming from your code that it is the former - in which case you have to purge the database first before adding the new entries to ensure there is no residue.