Translated into modern terms, it basically worked like ths: Suppose A and B fill out their tax returns, and A is assessed to pay more tax than B. Then A could sue B under the law of "antidosis" (the Greek work meant "exchange").
Now B has a choice: he can either pay the larger tax amount (and A will pay the smaller assessment). Or else B can claim that his smaller tax assessment is fair, because he has less money than A. In the latter case, a strange thing happens: A and B exchange all their property. Now A has to pay his original tax assessment, but out of B's (smaller) fortune.

Of course, if everyone is paying a fair amount of tax, A would be crazy to sue B, because B must be poorer than A (as he is assessed for less tax) so B would simply force the exchange and would end up with A's larger fortune.

Here's how it works in practice:
Suppose A has $1,000,000, B has $10,000,000 and C has just $10,000.
As A is richer than C, he pays more tax: perhaps C pays $100 but A pays $10,000.
But B is so rich that he can afford a clever accountant and only pays $5,000.

So A sues B. If B gives in, he will pay $10,000 tax and A will only pay $5,000.
But if B stands his ground, they will exchange property; B will still pay only $5,000 in tax, but out of his new possessions of $1,000,000, while A happily walks away with $10,000,000 to pay his larger tax bill.
So B's only sensible choice is to accept the larger tax bill.

So if B is to keep his fortune, he has to pay (willingly or otherwise) more tax than anyone who could stand to gain by suing him under this law - i.e. anyone with less money than himself.

The really ingenious thing is this: who is B afraid of? He is obviously not afraid of poor people like C, because it is easy for a rich person to ensure that he pays more tax than any poor person.
No, B's enemy in this situation is A: someone who is very rich, but not as rich as himself. Only other rich people are likely to be paying more tax than B and therefore to be in a position to sue.

So this law causes rich people to compete against each other to pay as much tax as possible; a cabal of ultra-rich people cannot conspire together to keep their own taxes low (as happens in our world) because any other rich person could force them to increase their tax payment.