In "Prisoner of Azkaban" chapter 20 ("The Dementor's Kiss"), Harry and Hermione follow Sirius to the lake, where they get surrounded by dementors. Harry's Patronus is unsuccessful, and one of the dementors lowers its hood and tries to Kiss Harry, before the dementors are finally driven away.
A couple chapters later, we get this exchange:
"And the dementors?" said Dumbledore. "They'll be removed from the school, I trust?"
"Oh yes, they'll have to go," said Fudge, running his fingers distractedly through his hair. "Never dreamed they'd attempt to administer the Kiss on an innocent boy... Completely out of control..."
Prisoner of Azkaban, chapter 22, "Owl Post Again"
How did Fudge know about the Kiss?
Sirius and Hermione both passed out before the dementor lowered its hood, so they didn't know it even happened.
We see all of Harry's dialogue between the lake and that exchange with Fudge, and Harry never mentioned the Kiss to anyone -- not Hermione, not Fudge, not even Dumbledore.
Snape hadn't come on the scene yet:
"What amazes me most is the behavior of the dementors... you've really no idea what made them retreat, Snape?"
"No, Minister... by the time I had come 'round they were heading back to their positions at the entrances..."
Prisoner of Azkaban, chapter 21, "Hermione's Secret"
The Kiss happened before the corporeal Patronus drove the dementors away. If Snape didn't see the Patronus, then he wasn't there in time to see the Kiss either.
The only other ones who were around were Lupin and Pettigrew. Lupin couldn't have told Fudge about it; he was in the forest at the time, probably in no state to observe, and definitely not in any position to communicate. And Pettigrew was hardly going to go talk to the Ministry.
We know dementors can communicate with wizards, so I suppose we can't rule out one of them telling the Minister what had happened after the fact. But it hardly seems they'd admit to being that far out of Ministry control.
So how did Fudge know about it?