<?php
// Check for empty fields
if(empty($_POST['name'])	||
   empty($_POST['email'])	||
   empty($_POST['phone'])	||
   empty($_POST['message'])	||
   !filter_var($_POST['email'], FILTER_VALIDATE_EMAIL))
   {
	echo "No arguments Provided!";
	return false;
   }
	
$name = strip_tags(htmlspecialchars($_POST['name']));
$email_address = strip_tags(htmlspecialchars($_POST['email']));
$phone = strip_tags(htmlspecialchars($_POST['phone']));
$message = strip_tags(htmlspecialchars($_POST['message']));
	
// Create the email and send the message
$to = 'yourname@yourdomain.com'; // Add your email address inbetween the '' replacing yourname@yourdomain.com - This is where the form will send a message to.
$email_subject = "Website Contact Form:  $name";
$email_body = "You have received a new message from your website contact form.\n\n"."Here are the details:\n\nName: $name\n\nEmail: $email_address\n\nPhone: $phone\n\nMessage:\n$message";
$headers = "From: noreply@yourdomain.com\n"; // This is the email address the generated message will be from. We recommend using something like noreply@yourdomain.com.
$headers .= "Reply-To: $email_address";	

return mail($to, $email_subject, $email_body, $headers);			
?>

<?php
// Check for empty fields
if(empty($_POST['name']) || empty($_POST['email']) || empty($_POST['phone']) || empty($_POST['message']) || !filter_var($_POST['email'], FILTER_VALIDATE_EMAIL)) {
  http_response_code(500);
  exit();
}

$name = strip_tags(htmlspecialchars($_POST['name']));
$email = strip_tags(htmlspecialchars($_POST['email']));
$phone = strip_tags(htmlspecialchars($_POST['phone']));
$message = strip_tags(htmlspecialchars($_POST['message']));

// Create the email and send the message
$to = "yourname@yourdomain.com"; // Add your email address inbetween the "" replacing yourname@yourdomain.com - This is where the form will send a message to.
$subject = "Website Contact Form:  $name";
$body = "You have received a new message from your website contact form.\n\n"."Here are the details:\n\nName: $name\n\nEmail: $email\n\nPhone: $phone\n\nMessage:\n$message";
$header = "From: noreply@yourdomain.com\n"; // This is the email address the generated message will be from. We recommend using something like noreply@yourdomain.com.
$header .= "Reply-To: $email";	

if(!mail($to, $subject, $body, $header))
  http_response_code(500);
?>

Use HTTP codes as response.
Use HTTP codes as response.

The code "$.ajax" in "contact_me.js" do not understand the "return true" or "return false" from PHP. It only understand the HTTP response code for select between "success" or "error".

This is a quick fix, would be better a JSON response.