The Romans of the West simply felt that the Empire, the one and only one to which they felt they belonged, had finally returned to free them from Germanic occupation.

The so-called 'fall of the Western Roman Empire' only became an important historical event from the 15th century onwards, well over a millennium after the deposition of Romulus Augustulus, when the new Renaissance Europeans felt the need to mark a break between the classical age they were inspired by and the Middle Ages from which they wanted to break away.

But for contemporaries it meant nothing in particular. First of all Romulus Augustulus was an illegitimate emperor put on the throne by a conspiracy. The true emperor recognised as an equal by the eastern emperor was Julius Nepot, who nominally continued to reign well after the end of the emperor-child. When Julius Nepot also died, there was no need for a new western emperor to rule over a territory that was now limited to the Italian province, which returned under the sovereignty of Constantinople.

In Vandal Africa, the Romans were indeed welcomed as liberators by the local Roman population. The Vandals were among the Germanic peoples who integrated least with the natives and never went beyond the role of raiders. And this was their downfall: to prevent the population from putting up an effective resistance, they destroyed all the city walls and forbade their reconstruction. When they were forced onto the defensive against Constantinople, they had no fortified city in which to take refuge.

The situation in Ostrogothic Italy was more complicated. Both Odoacer (the one who deposed Romulus Augustulus) and Theodoric ruled Italy in the name of Constantinople, with the recognition of the Roman emperor (the eastern one, the only one left) and with the support of the Roman Senate, still residing in Rome and still rich and influential. The gold coins struck in Italy had the face of the emperor of Constantinople and the coins in antiquity were the equivalent of a poster saying 'this person is in charge here'.

Obviously, Ostrogothic Italy, however loyal to the empire, was not equivalent to an imperial province. Taxes for example were withheld in Italy and the army of Italy was the army of the Goth king, allied to but not controlled by Constantinople. Justinian wanted Italy and Rome under his full control and chose to invade Italy in the belief that the Goths would fall just as easily as the Vandals, but instead of two years the war lasted twenty and the war between the Goths and Romans destroyed much of the Romanity of the peninsula, which had remained virtually intact until then.

To return to your question, the Italian population largely sided with the Empire, after all from their point of view Justinian was simply reasserting his legitimate control over Italy and the Goths were rebelling against an imperial order. The Senate in particular sided with Constantinople from the beginning and this marked its end: many fled to Greece and Asia Minor when the conflict showed that it would last for a long time, many were killed in revenge by the Goths. After the war, the Senate would no longer be an important player in Italian politics.

However, it is not correct to say that ALL Italian Romans welcomed the eastern armies. There were many pro-Goths even among the Latins. Several cities opened their gates to the Orientals or resisted them also depending on the presence or absence of Gothic troops within them. Note how in the very early years of the war the two main cities of peninsular Italy had opposite fates. Rome joyfully welcomed the arrival of General Belisarius and his eastern troops and then faced a harsh Goth siege. Naples, on the contrary, refused to let the Roman general enter and was besieged and then sacked by him. As the war went on, the clash also became a clash between social classes and the Goths received much support from the poorer parts of the Latin population, especially the slaves, when it became clear that an eventual Goth victory would mean the expropriation and exile of the landowning class.