All of your pages are located at http://www.yourcustomdomain.com by default, which you can see by going to the site in a fresh browser and clicking around.
The website domain only switches over to https://yoursite.leaguelab.com when the user logs in or registers because League Lab's SSL certificate is used to protect/encrypt all data transfer and verify the identity of the server. The certificate ensures that passwords and personal info are not sent in plain text across the internet for the protection of your users info and the security of site accounts (which includes your admin account).
This is the mechanism that puts the padlock in the address bar and the identity aspect of it is strongly tied to domain names. Our certificate can only be used on subdomains of leaguelab.com.