It’s not going to be automatic in this case. You will need to generate the private key and CSR yourself on your own computer, manually complete a domain verification step, and then send the files you created to Hostgator along with the certificate.
You can do this using a service like https://www.gethttpsforfree.com/ (which uses Let’s Encrypt). As the site says, it is intended for people who are already somewhat familiar with the process of obtaining a certificate using another certificate authority. You will need to get OpenSSL on your own computer (which you probably already have if you’re running Linux or Mac OS X, and probably don’t if you’re running Windows), and then run various command-line OpenSSL commands as directed by the gethttpsforfree site.
Since the certificate will expire in 90 days, you’ll also need to be prepared to repeat this process every 90 days (or a little less) in order to get a replacement certificate.
If that doesn’t sound like your cup of tea, you’re probably better off switching to a different hosting provider that has more automated Let’s Encrypt support, or waiting for Hostgator to provide its own Let’s Encrypt integration tools in the future.