The Three Lions took the lead against the Vatreni in Moscow on Wednesday but were pegged back to miss out on the chance to reach their first final since 1966.
Despite the heartbreak of the defeat, England are widely accepted to have exceeded expectations and still have the third-place playoff against Belgium to negotiate on Saturday.
In two years' time, Wembley Stadium will host several matches - including the semi-finals and final - as Euro 2020 takes place across various European cities.
"We almost have a home tournament, it's going to be brilliant," Southgate told Sky Sports News.
"What the players are going to experience is close to what we experienced in 1996 and in 1966. That's incredible for everybody.
"We did talk to [the players] about that before, but I think it was hard for them to really get it, most of them weren't born in 1990 and they were too young for '96.
"So they haven't felt what the nation really getting excited about the team feels like. They've seen that now."
The squad will return home to Birmingham on Sunday, but the Football Association has confirmed that there will be no parade despite holding one the last time England reached the semi-finals in 1990.