Put this after your if empty $errors and edit to meet your needs:
if(empty($errors)) {
$subject = __('[Your Subject]', 'your_text_domain').' '.$title;
$body = __("Name: ","your_text_domain").$booking_name;
$body .= __("\n\nEmail: ","your_text_domain").$booking_email;
$body .= __("\n\nPhone: ","your_text_domain").$booking_phone;
$body .= __("\n\nMessage: ","your_text_domain").$message;
$headers = __('From: [email protected] ', 'your_text_domain').' <'.$booking_email.'>' . "\r\n" . __('Reply-To: ','your_text_domain') . $email; // Change this $email to whatever you want where you will get messages from booked persons
$mail_sent = false;
$mail_sent = mail($mailto, $subject, $body, $headers);
if ($mail_sent == true) {
echo __('Your message has been sent succesfully.','your_text_domain');
} else {
$error = true;
echo __('Sorry, an error occured while sending your message. Please try again.','your_text_domain');
}
}