Similar to the login button, the presence of the logout button is conditionally dependent on the state of the amp-access component:

<button amp-access="loggedIn" amp-access-hide tabindex="0" on="tap:amp-access.login-sign-out" class="button-primary comment-button">Logout</button>

When you click the Logout button, you are directed to the URL that you specified in the amp-access JSON configuration, as part of the login object:

"login": {
  "sign-in": "",
  "sign-out": ""

Similar to the login, when the AMPByExample server receives a logout request, it uses the return URL query parameter automatically added by the AMP library and redirects to it, adding #success=true. By this time, you are back on the initial page; the AMPByExample cookie previously created for the login page (called ABE_LOGGED_IN) would be cleared at this point.