The Romans had a formal ritual, evocatio, whereby the god of an enemy people was ritually coaxed to defect from the enemy and then given cult in Rome. It seems quite rare, done for example with Juno of Veii, who was given a cult on the Aventine, and perhaps a similar evocatio was effected for the Juno of Carthage (i.e. Tanit). A minor evocatio is evidenced in Asia Minor in 75 BC, perhaps driven by the antiquarian lore.

There was a process for formally establishing new cults in the city of Rome: the Sibylline books would be consulted, and the Xviri sacris faciundis would advise the senate to engage in civic worship of a new god (generally including a new temple) in order to avert some calamity. This is how gods like Apollo (430) and Ascelpius (293) supposedly are brought to Rome, supposedly in response to plague. Mind you, this refers to the establishment of state cult--people were certainly privately worshiping these gods before a decision was made to sponsor a state cult. Similarly, the cult of Magna Mater, an Anatolian deity probably linked to Troy, was instituted in 204 BC, likely in hopes of aiding Rome in the Second Punic War.

At a personal level, Romans worshipping the gods of people they conquered was simply the result of good old-fashioned polytheistic syncretism. People lived in a world full of spirits, and worshipped gods they thought might be useful. If you are a merchant making a voyage across the English channel and you hear that a goddess named Nehalennia might keep you safe, they you make a vow to her, just to cover your bases. In the empire, worship of new gods was likely transmitted through social networks: for example Anna Collar Religious Networks in the Roman Empire traces the expansion of the Syrian god Jupiter Dolichenus through military camps across the Rhine and Danube: soldiers in Syria started worshipping him, and then spread the practice along their military network; soldiers and merchants seem to have been obvious vectors for religious innovation.