I have a GitHub page and I have a domain from namecheap. Now I've heard that Let's Encrypt offers free certificates.
Is it possible to make my site use HTTPS? How?
To make use of a certificate you not only need to have your own domain and certificate but you need to have access to the server configuration so that you actually could use the certificate. In case of github you don't have such deep access so you cannot make use of any certificate, no matter if you got it through Let's Encrypt or some other CA.