There are a number of reasons Roman emperors embarked on so many public works projects. The varied reasons can be put into two main categories. In general, they commissioned public works to enhance their own status and to help maintain stability and order in Rome.
Augustus, the first emperor of Rome, is alleged to have said something to the effect that Rome was a city of bricks when he came to power and a city of marble by the time his reign was drawing to a close. By saying this, Augustus was showing one reason for building—enhancing his status. Emperors had to try to show that they were powerful and that they had a vision for the future of Rome. By building public works, emperors could show this. They could show that they were able to amass the sort of wealth that was needed to build these works and they were able to show that they were trying to improve the city that was the center of their empire. Emperors also tried to improve their status by building religious and other buildings and by using their dedications as opportunities for propaganda. Augustus, for example, dedicated buildings in his own name but also in the names of members of his family. This helped to portray the emperor and his family as people who had the best interests of Rome in mind and/or who displayed piety and respect for the gods. By doing these things, emperors made themselves and those around them look better and helped make sure that they would be able to stay in power.
A second, though related, reason to build was to help maintain stability and order in Rome. One way this worked is alluded to in the previous paragraph. If the emperors could convince Romans that they were powerful and that they respected the gods, they made it less likely that ordinary Romans would be unhappy and want to rebel. Roman emperors could also use the public works as a way to try to instill the proper values in their subjects. They could build temples and monuments to famous heroes as a way to encourage Romans to venerate the gods and to buy in to the values that made Rome great. Again, this would encourage people to identify with the status quo and discourage them from wanting things to change. Finally, the building of public works could materially improve the lives of Romans. The building of aqueducts, for example, would clearly bring benefits to the ordinary people of Rome. Some of them would surely have been able to get work building the aqueducts. Still more of them would have benefitted from the water that came through the aqueducts. By engaging in projects like this, the emperors improved the material lives of the Romans and made them less likely to rebel.
We may think that emperors had unlimited and unchallenged power, but this is not so. They did not have to stand for election, but they did have to keep their subjects happy to some extent. By engaging in public works, they were (in part) trying to ensure that their subjects did not rebel against them.