During the SSL/TLS handshake process, the web browser and the website server exchange information to establish a secure connection. They communicate to verify the authenticity of site certificates and decide on the encryption or cipher suite to be used for secure communication. If there are discrepancies in this communication, such as incompatible SSL/TLS versions or cipher suites, the TLS handshake fails, leading to the "Err_SSL_Version_or_Cipher Mismatch" error.