This page is just a frontend that uses Let's Encrypt API to generate SSL. Visit How it works for a detailed explanation.
Let’s Encrypt is a nonprofit organization, on a mission to create a more secure and privacy respecting Web by promoting the widespread adoption of HTTPS.
The private key is always generated and managed on your own servers, not by Techlister or Let’s Encrypt certificate authority.
Let's Encrypt certificates are valid for 90 days. You can read about why here.