Did you push the "cPanel installation button"? If you don't have cPanel and/or the installation button doesn't work, you need to re-install the certificate manually into your webserver. Renewing is a two-step process: getting a (re)new(ed) certificate and making sure the webserver actually uses this new cert. Often this is an automatic process. But sometimes it's not.
Let's Encrypt recommends to renew after 2/3rds of the lifetime of the certificate. In the case of Let's Encrypt, that's after 60 days out of the 90 days lifetime of the cert.
So yes and no: yes, you need to renew the cert yourself (as you're using a manual process) and no, you shouldn't renew after the expiration date, but before.