How rich people get richer and the poors get poorer when the roman republic became an empire?

Inequality is a feature of any society (even the orthodox Communist ones that still exist) throughout history. It is moreover not true that simply because of the transition from Republic to Empire the rich got richer, and the poor poorer. It is however true that the acquisition of great chunks of new Roman-dominated soil (a greater part of that happened already in the late Republic years, however) gave Roman administrators of the conquered lands ample opportunity to enrich themselves. And the extension of the Roman Empire was a boon for trade and that again was also a source of new wealth.

But it was a very small group of men that became very rich because of that, and the growth of the Roman Republic and Empire also offered a lot of opportunities to others, so the poor did not by definition get poorer.

The only exception might have been the city of Rome itself where most of the rich and powerful lived, and where the 'proletariat' was not encouraged to find employment elsewhere - they and their votes were a power base for the rich and the politicians - but who increasingly became dependent on food and other handouts from the Government in Rome and from their rich patrons.