Speaking on Tuesday, Amfo said she had been so affected by Mr Floyd's death that she had missed her show on Monday.
"I didn't have the mental strength to face you guys yesterday," said the DJ, her voice breaking with emotion.
"I was sat on my sofa crying, angry, confused... stuck at the news of yet another brutalised black body."
Mr Floyd, a 46-year-old African-American man, died last week after a white police officer knelt on his neck for almost nine minutes.
Minneapolis policeman Derek Chauvin has been sacked and charged with third-degree murder.
Amfo was speaking on "Blackout Tuesday", an initiative demanding racial justice and structural change in the wake of the killing.
Originally organised by the music industry, it has involved stars like Rihanna, Lady Gaga and Drake observing a day of silence, and record labels suspending normal business for 24 hours.
The movement has now spread across social media, with many users posting a simple black square, alongside messages of solidarity and links to anti-racism resources.
Radio 1 and its sister station 1Xtra have been reflecting the movement by hosting discussions about the issues surrounding Mr Floyd's death, and playing songs that address black empowerment and identity.
Speaking on her mid-morning show, Amfo said the events in Minneapolis had reinforced a feeling among black people "that people want our culture, but they do not want us".
She added: "In other words, you want my talent, but you don't want me.
"There is a false idea that racism - and in this case anti-blackness - is just name-calling and physical violence, when it is so much more insidious than that.
"One of my favourite thinkers is a woman called Amanda Seales, and she says this and I feel it deeply when she says, 'You cannot enjoy the rhythm and ignore the blues'. And I say that with my chest."
The presenter ended her speech by playing Kendrick Lamar's Alright, which became associated with the Black Lives Matter movement after its release.
The song opens with the line: "All my life I had to fight," and references police officers who "wanna kill us dead in the street, for sure".
"Clara Amfo is just one of our finest and smartest broadcasters," wrote Pointless host Richard Osman. "She speaks to the Radio 1 audience, with great honesty, power and truth, about the murder of George Floyd."
All great changes are preceded by chaos.”