Depending on your plugin settings, the information on who created the post (contact record) is saved in the WordPress database. I believe what you're after to be able to see it in some specific place. At the moment it can be seen for example in the Edit Post screen under WP admin pages:
If you want to display it elsewhere, that's more of a theme than plugin related issue. Most themes display post authors by default. This of course requires that users have logged in before posting - otherwise there's nothing to differentiate between one anonymous submitter and another. If your users are not logged in when submitting, you could add a visible field on the page where they can write their own name before hitting the Submit button.