What language was mostly spoken in ancient Rome?

It was Greek, not Latin

A lingua franca is a language used between two people when neither is using their mother tongue. Rome was the capital city of a fast and expanding empire, a commercial hub of over a million people. 

Although the native Language of Rome (capital of Latium) was Latin, the lingua franca -the language you would use if you were buying or selling or generally trying to make yourself understood was koine or 'common' Greek. 

Greek was also the language of choice for Rome's educated urban elit. Sophisticated Romans saw themselves as the inheritors of Greek culture. Virgil's Aeneid -the epic poem that tells the story of Rome's foundation- makes it explicit that contemporary Rome grew directly out of the mythical Greece that Homer had written about. Speaking Greek at home was essential. Most of the literature that upper-class Romans read was in Greek; the art, architecture, horticulture, cookery and fashion they admired was Greek; and most of their teachers and domestic staff were Greeks. 

Even when they did speak Latin it wasn't the classical Latin that we recognise. For speaking native Romans used a form of the language called 'Vulgar Latin'. The word vurgar simply meant 'common' or 'of the people'. Classical Latin was the written language -used for law, oratory and administration but not for conversation. It was the everyday version that the Roman army carried across Europe and it was Vulgar, not classical Latin that spawned the Romance languages: Italian, French and Spanish.

But Vulgar Latin was only the daily language of Latium, not the Empire. Greek was the first language of the eastern Empire, based around Constantinople and of the cities in southern Italy. The name Naples (Neapolis in Latin) is actually Greek (nea, new, polis, city). Today, the local dialect in Naples, Neopolitana, still shows traces of Greek and the Griko language is still spoken by 30,000 in southern Italy. Modern Greek and Griko are close enough for speakers to be able to understand one another. Greek, not Latin, was the popular choice for the Mediterranean marketplace.

Lingua Franca was originally an Italian -not a Latin- term for the specific language that was used by people trading in the Mediterranean from the eleventh to the nineteenth century. Based on Italian it combined elements of Provencal, Spanish, Portuguese, Greek, French and Arabicinto a flexible lingo everyone could speak and understand. 

Lingua Franca doesn't mean 'French language', but 'Language of the Franks'. It derives from the Arabic habit of referring to all Christians as 'Franks' (rather as we once referred to all Muslims as 'Moors'). Franji remains a common Arabic word used to describe Westerners today. 

source: John Lloyd & John Mitchinson, The Second Book of General Ignorance, p. 119-120
.
~ Για ελληνικά:
Γιατί οι Ρωμαίοι μιλούσαν ελληνικά, είχαν παιδαγωγούς Έλληνες και θεωρούσαν ότι είναι κληρονόμοι του αρχαιοελληνικού πολιτισμού