Klopp made the announcement well before the end of the season.
"It was a really strange situation," Elliott told Amazon Prime video.
"Jurgen wasn't really one for having meetings after a game. All the lads were speaking saying 'What's going on?' because it's never happened before.
"As he walked in, you knew something was wrong and that he was going to say something because he is normally bubbly and joking around. That's when he told us.
"It was really weird that day. It felt like you lost someone. Everyone was just quiet. It felt like the whole building was quiet because we didn't know how to take the news.
"Up until he left it was strange, trying to get your head around the fact he was leaving."
Elliott also paid tribute to Klopp, stating the club's spirit has been revived under him.
"The players, the fans, everyone in this building and around the world feels together and I think that's because of him," Elliott said.
"I'm surprised there isn't a statue of him yet. He came in, he built the team up. He won so many things, he has a legacy that will live for ever. Even after I'm dead, it will go on.
"He deserves it for what he has done for this team, made it like a family. The way he is as a person, the way he handles everything that comes his way. He is a legend in my eyes and will be a legend at this club for ever."