His side finished top of Group C on Tuesday after a goalless draw against Slovenia followed a 1-1 stalemate with Denmark in their previous group-stage game.
Southgate’s side were booed off, and some empty cups were thrown at the Three Lions boss following another below-par display against a team ranked 57th in the world, 52 places lower than England.
“I’m not going to back away from it,” he said. “The most important thing is the supporters stay with the team.”
Southgate said he “understands” why fans have expressed their displeasure at disappointing performances and results, which included a 1-0 success over 32nd-ranked Serbia in their opening match of the tournament.
Fans jeered England at half-time in Cologne and Southgate was met by another negative response as he went to applaud supporters at full-time, with a handful of plastic pint cups thrown in his direction.
“I understand the narrative towards me and that’s better for the team than it being towards them,” he added.
“But it is creating an unusual environment to operate in. I’ve not seen any other team qualify and receive similar. I understand it, I’m not going to back away from it, but I’m very proud of the players for how they’re operating within it.”
England’s performance against Denmark in their second group game was widely criticised by pundits, with captain Harry Kane calling for more positivity from the media in the build-up to Slovenia, and saying ex-players should remember how “tough” tournament football is.
Social media video from inside Cologne Stadium appeared to show some England fans swearing towards Southgate, then arguing with other supporters who felt they should be getting behind the team and the manager.
Southgate gave several post-match interviews to individual broadcasters before holding a news conference.
He told beIN Sports that “the reaction to everything” was creating “an incredible environment” and the “players are feeling that”.
“I’ve lived in this environment for 20 years as a player and coach. We’ve got to keep ourselves on track and make sure the dressing room is tight.”
In his news conference later, Southgate added: “We have made England over the last three or four years fun again. I think it has been enjoyable for the players and we have got to be very, very careful that it stays that way.”
Southgate also revealed the responses from some supporters are “causing an issue for the group” and urged them to back the players.
He told BBC Radio 5 Live: “I was not going to back away from going over to thank people for coming and giving the support that they did, but I know that this is causing an issue for the group.
“I can deal with that but I need them to support the players.”