Rashford was the surprise selection in Roy Hodgson's squad after a rapid rise from United's youth team to first-team regular, having only made his debut in February.
"When I first went into the changing room and saw all of the players' names on the back of their shirts, it didn't seem real," Rashford said in an interview alongside club and national captain Wayne Rooney on the Football Association's website on Thursday.
"I remember going home after and speaking to my brother about it, and he agreed with me. He didn't think it was real either. That's what it's like until you get used to it."
Rashford marked his Manchester United debut with two goals against Denmark's Midtjylland in the Europa League in February, then scored two more in his Premier League debut against Arsenal.
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He scored again on his England debut in the first few minutes of a friendly against Australia last month.
"(At Christmas) I was actually trying to work my way into (United's) under-21s side and all of a sudden I appeared in the first team. You have to be ready when your chance comes," he said.
"At the end of the season you hear talk and stuff, but to be here now is just amazing. For it to be in my hands to go out and help the country is an amazing feeling."
Rashford is vying with Harry Kane, Jamie Vardy, Rooney and Daniel Sturridge for a striking berth.
"I think it's a good way to end the season for me, and hopefully we get some joy in the tournament," he said.
(Reporting by Martyn Herman. Editing by Adrian Warner.)
