The full quote from the Daily Prophet article is as follows:
Arnold Weasely, who was charged with possession of a flying car two years ago, was yesterday involved in tussle with several Muggle law-keepers ("policemen") over a number of highly aggressive dustbins. Mr. Weasley appears to have rushed to the aid of "Mad-Eye" Moody, the aged ex-Auror who retired from the Ministry when no longer able to tell the difference between a handshake and an attempted murder. Unsurprisingly, Mr. Weasely found, upon arrival at the Mr. Moody's heavily guarded house, that Mr. Moody had once again raised a false alarm. Mr. Weasely was forced to modify several memories before he could escape from the policemen, but refused to answer Daily Prophet questions about why he had involved the Ministry in such an undignified and potentially embarrassing scene.
To understand this article, we first need to go back earlier in the book, when Arthur (the Daily Prophet got his name wrong to boot) was called to help Moody:
Amos Diggory’s head was sitting in the middle of the flames like a large, bearded egg. It was talking very fast, completely unperturbed by the sparks flying around it and the flames licking its ears. “...Muggle neighbors heard bangs and shouting, so they went and called those what-d’you-call-’ems — please-men. Arthur, you’ve got to get over there —”
“Here!” said Mrs. Weasley breathlessly, pushing a piece of parchment, a bottle of ink, and a crumpled quill into Mr. Weasley’s hands.
“— it’s a real stroke of luck I heard about it,” said Mr. Diggory’s head. “I had to come into the office early to send a couple of owls, and I found the Improper Use of Magic lot all setting off — if Rita Skeeter gets hold of this one, Arthur —”
“What does Mad-Eye say happened?” asked Mr. Weasley, unscrewing the ink bottle, loading up his quill, and preparing to take notes.
Mr. Diggory’s head rolled its eyes. “Says he heard an intruder in his yard. Says he was creeping toward the house, but was ambushed by his dustbins.”
“What did the dustbins do?” asked Mr. Weasley, scribbling frantically.
“Made one hell of a noise and fired rubbish everywhere, as far as I can tell,” said Mr. Diggory. “Apparently one of them was still rocketing around when the please-men turned up —”
Mr. Weasley groaned. “And what about the intruder?”
“Arthur, you know Mad-Eye,” said Mr. Diggory’s head, rolling its eyes again. “Someone creeping into his yard in the dead of night? More likely there’s a very shell-shocked cat wandering around somewhere, covered in potato peelings. But if the Improper Use of Magic lot get their hands on Mad-Eye, he’s had it — think of his record — we’ve got to get him off on a minor charge, something in your department — what are exploding dustbins worth?”
“Might be a caution,” said Mr. Weasley, still writing very fast, his brow furrowed. “Mad-Eye didn’t use his wand? He didn’t actually attack anyone?”
“I’ll bet he leapt out of bed and started jinxing everything he
could reach through the window,” said Mr. Diggory, “but they’ll have a job proving it, there aren’t any casualties.”
“All right, I’m off,” Mr. Weasley said, and he stuffed the parchment with his notes on it into his pocket and dashed out of the kitchen again
This incident is how Mr. Weasely got involved. There was some sort of commotion at Mad-Eye Moody'd residence, causing Muggle neighbors to call the police. Arthur was called in to try to help Moody avoid a larger sentence (apparently the first Ministry representative on the scene has jurisdiction and can decide all charges, in the Wizard world?). The "false claim" was that of Moody claiming he had been attacked, when in fact there was no intruder at all.
Of course, later in the book we learn that
The commotion was actually Barty Crouch Jr. attacking and kidnapping the real Mad-Eye Moody. The story about the dustbins was something Crouch Jr. made up, a story he felt the wizarding world would believe the real Moody might actually say, in order to explain away the commotion that Moody's muggle neighbors heard in the first place.