England, who reached the last four of the World Cup in 2018, have got to the Euro semis only once in the last 50 years - in 1996 when Southgate was playing for the country.
"The opportunity is for us to go further than so many teams in the past and our European Championship record has been appalling," Southgate told reporters.
"They've got the opportunity to really excite our country again.
They know what that felt like two years ago from a slightly further distance and this time they get the chance to feel it ... really close up.
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England will face Croatia, the Czech Republic and either Scotland, Israel, Norway or Serbia in Group D. They open their campaign against Croatia on June 14.
England will play all three of their group games at Wembley, which will also host the semi-finals and final.