Luke called to her for help and she briefly tapped into the Force to help find him. They didn't know where Luke was till she did that and it was not explained in ESB. I always interpreted it as she didn't understand what was happening but just knew.
Vader said it, but no one knew for sure at the end of the film if it was true. Luke wasn't sure till Yoda confirmed it in ROTJ. That scene of Yoda telling Luke it was true was meant just as much to confirm it to the audience as it was to Luke. I assure you many fans debated of Vader might be lying or not until ROTJ came out. It was stated in exposition as you said, yet was still a plot thread that needed to be fully resolved.
Edited by Jr Mouse - 1/8/16 at 5:04pm