SPQR is a latin acronym: "Senatus Populusque Romanus". It means "The Senate and People of Rome" and was, as /u/lux_roth_chop said, an expression of the Roman view of the world.

For the first centuries of Rome, until its death as a republic around the turn of the era, the Roman identity was based upon their democratic, self-government. The Senate was the ruling body of aristocrats which voted on the day to day running of the city-state, and the People were split into various Assemblies so they could vote on matters of particular import. Of course, the system was inherently flawed and designed to keep the rich Senatorial families in power, and for the society to remain an Oligarchy rather than a proper Democracy. However, this form of government was central to the Roman identity.

They hated the very idea of kingship or autocracy, and this was reflected in their legends and histories, notably the story of Cincinnatus. Cincinnatus was a Roman aristocrat who, in a time of great need during war, was nominated as dictator, and legally had full control of Rome and her people for a period of six months. However, he defeated Rome's enemies in just 16 days and after that, promptly gave up his post of supreme power and returned to his farm, becoming a legend of civic virtue.

This attitude of service to the state before oneself was central to the beliefs of the Roman people, and because they were so proud of their method of government, stamped SPQR all over the place: coins, banners, civic buildings, documents, triumphal arches, the works. It was a way of saying "this object was created for the good of the Senate and People of Rome" and was intended to reinforce the idea that Rome was a city run by her people, for the good of her people.