Basically the answer is very simple: Rome was not a monolingual empire, even at state level. Learned people in rome, politicians, nobles, scholars, even high ranking army officers would be educated to speak both Latin and Greek. The entire eastern mediterranean had started using greek as a lingua franca after Alexander's conquests, and Greek also happened to be a "prestige language" among the Roman elite, so they never made any effort to dislodge or replace it. They simply adopted it as the secondary state language, and thus Greek continued it's dominance in the east.

As for why the Romans were so accepting of Greek to begin with, the answer to that is also fairly simple: The greeks colonized southern Italy long before Rome emerged as a major power, and thus they'd been in constant contact with the Greek language and culture since they were a tiny city state.