The basic answer to this is that the Roman Republic failed because of the changes that occurred as Rome came to control a huge empire. During this time, Rome became less egalitarian. Secondly, its finances were suffering. Finally, its military was becoming too strong. All of these things helped destroy the Roman Republic and make Rome’s experience with a republic a failure.
When Rome won a large empire, it became less egalitarian. When Rome was smaller, it was made up largely of citizens who were fairly equal to one another. However, as it became larger, its economy came to be dominated by large landowners who owned many slaves taken in the wars of empire. These people became rich and dominant, outcompeting the small farmers and causing them to lose their lands. This meant that a large number of citizens were now poor and living in the city of Rome. They were unhappy and their anger destabilized Rome.
As Rome got bigger, its government needed more and more money. It was not easy for the government to get all of the money that it needed in taxes. The government tried to get more and more taxes from the people. This, of course, made the people angry and their anger too helped to destabilize the Roman Republic.
Finally, the military of Rome was becoming stronger. The military was becoming more and more important because it was needed to protect and expand the empire. When the military got stronger, its leaders also became more important. The soldiers were generally willing to do whatever their generals said because they depended on those generals for wealth. The generals came to feel that they were more important than the government and they were, at times, willing to use their power to go against the government. As more military leaders rebelled against the Republic, it became weaker and weaker.
With all these problems, the Republic was unable to survive. Eventually, one of the military leaders, Julius Caesar, took power in Rome and essentially overthrew the Republic. After that, the Republic was dead and Rome was ruled by emperors.
While tradition and some semblance of power would remain, the foundation of government under a single figure was a requirement to continue the advancement of the Empire.