“I feel invisible a lot of the time,” Katie Kuiper tells PEOPLE

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PHOTO: CHRIS JACKSON/GETTY

When Katie Kuiper got a kiss from Prince Harry at the Invictus Games on May 9, it was a moment she could never have predicted – and one she will never forget.

After medaling in cycling, “We had been chatting with Prince Harry and he was hanging out,” Kuiper, 33, tells PEOPLE.

“I’m kind of shy, but I was like, ‘Do you mind if we get a photo of just the two of us?’ ”

He happily agreed, and gave her a kiss on the cheek.

“It was really special for me,” says Kuiper of San Antonio, Texas, who took home three golds and one silver in track and field and cycling events. “I feel invisible a lot of the time because of the way I look. He made me feel special.”

Kuiper, a former Army staff sergeant, served in Iraq and Guantanamo Bay Naval Base. Three years ago, she suffered a self-inflicted gunshot wound to her face.

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“People always ask me why, but I don’t know,” she says. “It was a lot of things building.” Asked if she struggled before her service, she said , “Oh, no.”

In his recent interview with PEOPLE, Prince Harry spoke about the “invisible injuries” of war and his mission to help those suffering with mental health problems in silence.

“It’s something that people talk about a little bit but need to understand a hell of a lot more,” Harry told PEOPLE. “But it is not something that just affects servicemen and women.”

“If you have a sprained ankle, you go to [the emergency room]. There needs to be something for mental health.”

The Invictus Games, says Kuiper, “are really important. They are really making a difference in people’s lives.”

Earlier in the day, Taryn Nixon, whose husband Adam represented the U.K. wheelchair basketball team, told PEOPLE of Prince Harry, “He’s literally brought people back from the brink,” she says. “They’ve got something to smile for, live for. They have passion and drive that they did not have before.

“Prince Harry is absolutely phenomenal,” she adds. “He’s a legend.”

Added her 13-year-old daughter Abi: “Invictus inspired my dad to get up. We have so many family days out now and spend so much time together because of the Games. Because of Prince Harry. It’s brought our family closer together.”