By default there is one checkbox called 'Send yourself a copy' is on contact form which is by default unchecked. As i can see it is not visible in attached screenshot. Maybe it hidden because of captcha.
Or else we can alter form using hook_form_alter on .module file and add following code.
if ($form['#form_id'] == 'contact_site_form') {
$form['copy']['#access'] = FALSE;
}