"My dad is so funny. He happened to be right backstage, standing right next to Adele before she went on to perform," the soulful breakout – who's partnering with Coca-Cola and McDonald's for the Rise Up Essence Festival program – tells PEOPLE.
"My parents are not in the business, so they have no sort of concept of: 'Is this appropriate?' They were just sort of hanging out, so my dad walks up to Adele and was like, 'Hey, you know, my daughter is Andra Day.' And of course she's so sweet! She obliges him. She starts giving a massage on his back," Day, 31, says.
Feigning a British accent, Day says Adele "was like, 'Oh my God, you must be so proud! This is beautiful and powerful!' And he's like, 'Have a good show!' and she goes on. When I heard that story, I just cried – I laughed so hard."
At the awards – where she was up for Best R&B Performance and Best R&B Album – Day teamed up with Ellie Goulding for a mash-up of their hits "Rise Up" and "Love Me Like You Do." And the singer says it's a blessing she made it through the set.
"Really, having a voice that night on the Grammys was my favorite part [because] the day before, I caught strep throat," she says, adding that her British collaborator proved a solid support system. "I could not have picked a better partner for that day. She's incredible and so sweet and supportive."
A West Coast native, Day hustled for 15 years before catching her big break, releasing her critically acclaimed debut, Cheers to the Fall, last year.
In addition to her faith and tight-knit family, Day (née Cassandra Monique Batie) cites both R&B icons and contemporaries as her motivators.
"Lauryn Hill is one of them [I admire], Erykah Badu, Jill Scott: I see them as very strong women who own who they are," she says. "And women like Billie Holiday and Nina Simone because they just were able to convey emotion and were very vulnerable and raw about it, which speaks to me in music."
Now with a far-reaching voice of her own, Day hopes to inspires others, starting with the Coke, McDonald's and Essence Fest collaboration.
"It's a huge platform, so to use it to encourage women and highlight women who are trailblazer, I'm so happy to be a part of it," she says.
Day's face and song lyrics will appear on Coke cups at McDonald's starting Monday in a campaign to inspire women to "Rise Up."
The singer will perform at Essence Fest for the second year in a row in New Orleans over the Fourth of July weekend.