Easy and best solution is the following.
index.html
<form action="submit.php" method="POST">
<input type="text" name="name" value="" />
<input type="text" name="email" value="" />
<textarea type="text" name="message"></textarea>
<div class="g-recaptcha" data-sitekey="Insert Your Site Key"></div>
<input type="submit" name="submit" value="SUBMIT">
</form>
submit.php
<?php
if(isset($_POST['submit']) && !empty($_POST['submit'])){
if(isset($_POST['g-recaptcha-response']) && !empty($_POST['g-recaptcha-response'])){
//your site secret key
$secret = 'InsertSiteSecretKey';
//get verify response data
$verifyResponse = file_get_contents('https://www.google.com/recaptcha/api/siteverify?secret='.$secret.'&response='.$_POST['g-recaptcha-response']);
$responseData = json_decode($verifyResponse);
if($responseData->success){
//contact form submission code goes here
$succMsg = 'Your contact request have submitted successfully.';
}else{
$errMsg = 'Robot verification failed, please try again.';
}
}else{
$errMsg = 'Please click on the reCAPTCHA box.';
}
}
?>
I have found this reference and full tutorial from here - Using new Google reCAPTCHA with PHP