This is a great question; there were many institutions that the Romans used to spread their culture. Let me name a few of them.
First, the Romans possessed one of the greatest military powers in the history of the world. This fact alone spread Roman society. They defeated people after people. For example, in less than 100 years, they conquered all the Hellenistic kingdoms along with Carthage. This point must not be under-emphasized.
Second, the Romans prized the art of government. They believed that their "gift" from the gods was the ability to rule. The poet, Virgil, has Jupiter say of the Romans that he has given them an empire without end. From a more practical point of view, Polybius, the Greek historian of Rome, makes much of the Roman constitution. He argues that they had a mixed constitution that enabled them to rule over a vast empire. For example, Rome had a senate, consuls, praetors, quaestors, tribunes, and so many other officials that could govern effectively.
Third, the Romans had the most advanced law in the ancient world. It is arguable that the greatest achievement in Roman civilization were the law codes. Even today by our standards we can say that Justinian's Digest is a massive accomplishment.
Finally, the Romans possessed great engineering. Hence, they built roads, bridges, aqueducts, and many other structures to create infrastructure.