Comment 3 for bug 759619

Banshee doesn't ask for your Last.fm password because that's the way the Last.fm authentication process works.
This is more secure, as you don't have to trust your Last.fm password to third-party applications and those applications don't have to store it.

When you see the "You need to allow Banshee to access your Last.fm account", Banshee should have opened a page on the Last.fm web site in your browser. It is in this web page that you have to authorize Banshee to access your Last.fm account. You might also have to log in to the Last.fm website.

The web page opened by Banshee might not be visible, if for example you already have a browser window open but not visible, it will just open a new tab in that window. Please check whether it is the case.