The claim from the website TV Tropes:
,”The Roman Republic enshrined this principle in its basic law, the Twelve Tables, which imposed a fixed fine for common assault, payable to the person who was assaulted. (The fine was a large one, but regrettably, the Romans never quite got around to adjusting it for inflation. By the time of Julius Caesar, repeated debasements of the currency had rendered the fine almost worthless, and rich sociopaths used to walk down the streets punching people in the face and immediately handing over the small bag of copper coins that, by law, was all the compensation their victims were entitled to.)”
So there are definite elements of truth here, but it's a little misleading.
The Twelve Tables did specify fixed fines for various delicts that we would categorize as assault (called iniuria), and the fine for the least serious assaults was 25 bronze coins (called asses, sing. as). Due to debasement of the currency, this amount would have become worth less and less over time. However, in the middle republic, the praetors issued a series of edicts allowing for a cause of action (actio in factum) in which plaintiffs could allege specific harms and damages instead of being bound by the fixed fines, effectively eliminating the fixed fines.
What precipitated the change is not necessarily clear, but Aulus Gellius relates a story of one Lucius Veratius who "'was an exceedingly wicked man and of cruel brutality. He used to amuse himself by striking free men in the face with his open hand. A slave followed him with a purse full of asses; as often as he had buffeted anyone, he ordered twenty-five asses to be counted out at once, according to the provision of the Twelve Tables. Therefore,' he continued, 'the praetors afterwards decided that this law was obsolete and invalid and declared that they would appoint arbiters to appraise damages.'" from Gel. 20.1
Aulus Gellius attributes this to the jurist Labeo, whose works do not survive independently of these kinds of quotations, but who was widely cited by later surviving jurists. The passage in Gellius is (to my knowledge) the only source for the anecdote of Veratius. The jurists whose works survive merely note the fixed fines, the effect of inflation, and the praetorian revisions (see e.g. Gaius 3.223).
It's entirely possible that the Veratius story was a historical event that prompted the praetorian edicts, but it may have been a retrojection by later generations to explain an incongruity between the 12 tables and the subsequent state of the law.
So the general contours of the claim are correct, but the false/misleading parts are that
Sources:
Borkowski's Textbook on Roman Law
Aulus Gellius's Attic Nights
The Institutes of Gaius
There's only one reference to this occurring. In his Attic Nights, the second-century antiquarian Aulus Gellius mentions a man of supple wrists and vestigial morals who ran around Rome slapping commoners:
"Lucius Veratius was an exceedingly wicked man and of cruel brutality. He used to amuse himself by striking free men in the face with his open hand. A slave followed him with a purse full of asses [low-value coins]; as often as he had buffeted anyone, he ordered twenty-five asses to be counted out at once, according to the provision of the Twelve Tables." (20.1.13)
As far as we know, nobody else ever abused the law or the faces of Roman commoners this way. The loophole that Veratius exploited was soon closed.