When you log on Windows creates an access token. This identifies you, the groups you are a member of and your privileges. And note that whether a user is an administrator or not is determined by whether the user is a member of the Administrators group.
Without UAC, when you run a program it gets a copy of the access token, and this controls what the program can access.
With UAC, when you run a program it gets a restricted access token. This is the original access token with "Administrators" removed from the list of groups (and some other changes). Even though your user is a member of the Administrators group, the program can't use Administrator privileges.
When you select "Run as Administrator" and your user is an administrator the program is launched with the original unrestricted access token. If your user is not an administrator you are prompted for an administrator account, and the program is run under that account.