I can access access_token, but now I am sitting at:
https://graph.facebook.com/oauth/access_token?client_id=ID&redirect_uri=http://www.EXAMPLE.com/Home/FBTempIndex/&client_secret=SECRET&code=CODE-CODE-CODE#_=_
and On this page, I can see:
access_token=ACTUAL-ACCESS-TOKEN&expires=5130106
That’s all. I am no longer accessing this page from my server. How can I do that?
My requests end at FB servers. I confirmed it via Google Analytics and Fiddler. Here’s fiddler image:

for better understanding, how can I request GET me?acc... when user is no longer on my website?

You take that access_token you get back then use it to query the Graph API.
Or am I missing something in your question?