Kanan, Ezra, and Ahsoka make a connection through the Force to impede Darth Vader, and to find out who he is. It is at this point that Ahsoka learns that he was once Anakin Skywalker, causing her to panic, to scream out “No!”, and break the Force connection, which causes her to lose consciousness for the remainder of the battle.
Darth Vader, conversely, learns who Ahsoka is, even saying “So, the apprentice lives!” in response to learning of her survival. (It should be noted that some have misheard the line as “Hello, Snips.” due to the applause that occurred in the audience when Ahsoka learned of the truth. This is not what he says, hilariously out-of-character though it would have been. He also does not say “My apprentice lives!” as some have reported for a reason that will be detailed below.) Darth Vader then makes it his mission to bring Ahsoka and the Ghost crew back alive for his own purposes.