The anime is .Hack//ROOTS
In 2015, CC Corporation's datacenter caught on fire destroying "The World". By splicing in data from what would have been another project, CC Corp. was able to rebuild the game. In 2016, "The World R:2" was released. .hack//Roots covers the time Haseo spent with Shino and Ovan in the guild known as the Twilight Brigade, before the .hack//G.U. PS2 games.
as per the description, it's the prequel to the PS2 game Series .Hack//G.U. particular it goes though the events that occur between the scenes of the intro to the first game (from when Haseo is first saved by Ovan until his fight, defeat and re-initialization at the hands of "Tri-Edge")
the scene you refer to is Episode 17 - Painful Forest and Episode 18 - Limit
Set during the Forest of Pain event players are tasked to reach the deepest part of the forest. upon reaching it Taihaku (the one who finds the Greatsword) and Haseo (who finds a pair of Twin Blades) are teleported to another area outside the game
Within this area each one encounters a red figure who asks if they have seen it's Daughter. those familiar with the previous games/series will recognize this has Harald Hoerwick, the one who first developed the Beta of The World called Fragment and the "Daughter" he asks about is Aura, the AI Child he created in the previous games
Taihaku answers that he does not know Harald's daughter but wishes that she is ok to which Harald bestows onto him the sword he found. Haseo, consumed by rage in his quest to destroy "Tri-Edge" who turned Shino into a Last One answers he doesn't care about Harald's Daughter and demands to know where "Tri-Edge" is. Harald refuses to give him the weapon and Haseo is struck by something.
before the Forest of Pain is deleted Tabby is waiting for Haseo to return and when he does she finds that he no longer looks the same. Haseo's Class, the Adept Rouge, can be powered up by obtaining new Forms and after what struck him Haseo has gone from the original 1st Form to the 3rd Form
i can't remember the woman on a computer but i assume this would be
Ender/Pai - Real name is Reiko Saeki, a Member of the Guild, Raven, who's members are apart of Project G.U and are Epitaph Users who wield Avatars, the manifestation of one of the 8 Phases of Morganna
i believe is her as at the time the only other female character is Tabby however i don't recall her talking a man outside the game and i believe the man is
Naobi/Yata - Real Name is Takumi Hino and is the .hacker Wiseman from the previous series of games. he is the leader of Raven and assists Reiko as they both work for CC Corp while he has deeper knowledge in the original Project G.U which Reiko's missing brother took part in