How to update server side reCaptcha PHP - php

Question

I am fairly new to bootstrap and this is the first I have used reCaptcha. I also don't have a lot of experience with PHP. I have designed numerous web sites and am very familiar with html(5) and css(3).
I can easily add the reCaptcha widget to my form and it displays correctly. However, I have no clue how to update my PHP file for the server side processing. Below is the code for my bootstrap contact form section and the code for my PHP file.
HTML (contact section):
<!-- Contact Section -->
<section id="contact">
<div class="container">
<div class="row">
<div class="col-lg-12 text-center">
<h2>Contact Me</h2>
<hr class="star-primary">
</div>
</div>
<div class="row">
<div class="col-lg-8 col-lg-offset-2">
<form name="sentMessage" id="contactForm" novalidate>
<div class="row control-group">
<div class="form-group col-xs-12 floating-label-form-group controls">
<label>Name</label>
<input type="text" class="form-control" placeholder="Name" id="name" required data-validation-required-message="Please enter your name.">
<p class="help-block text-danger"></p>
</div>
</div>
<div class="row control-group">
<div class="form-group col-xs-12 floating-label-form-group controls">
<label>Email Address</label>
<input type="email" class="form-control" placeholder="Email Address" id="email" required data-validation-required-message="Please enter your email address.">
<p class="help-block text-danger"></p>
</div>
</div>
<div class="row control-group">
<div class="form-group col-xs-12 floating-label-form-group controls">
<label>Phone Number</label>
<input type="tel" class="form-control" placeholder="Phone Number" id="phone" required data-validation-required-message="Please enter your phone number.">
<p class="help-block text-danger"></p>
</div>
</div>
<div class="row control-group">
<div class="form-group col-xs-12 floating-label-form-group controls">
<label>Message</label>
<textarea rows="5" class="form-control" placeholder="Message" id="message" required data-validation-required-message="Please enter a message."></textarea>
<p class="help-block text-danger"></p>
</div>
</div>
<br>
<div class="g-recaptcha" data-sitekey="mySitekey"></div>
<br>
<div id="success"></div>
<div class="row">
<div class="form-group col-xs-12">
<button type="submit" class="btn btn-success btn-lg">Send</button>
</div>
</div>
</form>
</div>
</div>
</div>
</section>
PHP (where do I put the required server-side php code here?):
<?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 = $_POST['name'];
$email_address = $_POST['email'];
$phone = $_POST['phone'];
$message = $_POST['message'];
// Create the email and send the message
$to = 'myEmail'; // This is where the form will send a message to.
$email_subject = "Portfolio Website Message: $name";
$email_body = "You have received a new message from your portfolio 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#mydomain.com\n"; // This is the email address the generated message will be from.
$headers .= "Reply-To: $email_address";
mail($to,$email_subject,$email_body,$headers);
return true;
?>
Thanks for your help!