I'm not familiar with the package you're using, nor with how Facebook manages this, but there are a couple of ways in which is could be accomplished.
Facebook, requiring you to login, presumably maintains a database containing details of your posts, what you've read, and thos items to which you've 'liked'. This seems a bit heavy for your site, which I notice doesn't require a login.
A simpler method may be to store the 'like' information in a cookie. This is a small file that gets written to the user's hard drive and can contain all manner of infirmation about that user's activities on the site, such as the last time they logged-on. I don't see why it couldn't also contain information that would support your 'like' feature. Of course you'd need a mechanism for re-reading the cookie (when they visit your site) and interpretting the information it contains but that's a bit complicated for here.
I'd recommend getting hold of a couple of good books on the subject of html and website design.