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It was the first time the Midlands club had let in six at home in a league game since October 1983.
Villa remain bottom of the Premier League and eight points from safety.
"I feel humiliated. I am sorry for the fans who came to Villa Park," said Garde. "This is a bad feeling that we did not fight enough."
Villa, owned by American Randy Lerner, let in two goals before half-time and then four during a 13-minute spell after the break.
The fans began leaving during the second half, while some home supporters also chanted 'Lerner out'.
"Before the game you could expect the level of the two teams was not on the same one but, after quite a good 15 minutes, I don't know what happened," said Garde.
"The first two goals were difficult to digest and then my team did not play anymore.
"We were always two or three metres away from from the opponent. It was very easy for them.
"I was expecting fighting spirit, especially in the position we are in, but we were not able to deliver it."
Garde, who was appointed Villa boss in November on a three-and-a-half-year deal, refused to concede his side were doomed for the drop into the Championship.
"I was not expecting this kind of match and we have to go for the next one to win it," said the Frenchman.
"Until we have little hope, I will be the one who will still believe. After a game like this, it is very difficult but we won't play Liverpool every weekend.
"I understand [the fans' frustration]. I don't know exactly who let down who in this football club.
"I am trying to concentrate on my job and try to do the best with the players who want to still fight."
BBC Radio 5 live's Pat Murphy: "Villa were an absolute shambles. The biggest round of applause came from the Villa fans when former striker Christian Benteke came on. I think the fans know that he did more than anyone to keep Villa up in the last few seasons.
"The Villa players just aren't good enough. There was a derisory cheer when Scott Sinclair hit the post. What must the European Cup winners from a proud Aston Villa side think of this?
"I don't blame Remi Garde but after this game it is 15 games and 12 points. He got a hospital pass.
"He is not happy at all about the lack of investment. It will take a lot to make him stay on. What is in it for him, after doing so much in Ligue 1?"
Aston Villa manager Remi Garde said he felt "humiliated" and apologised for his side's performance in the 6-0 thrashing by Liverpool.