Why was the roman empire crowned by Charlemagne?

I think the question you are asking is why Charlemagne was crowned emperor of the Roman Empire. Though we do not have a really good record of why this happened, it is probable that the people involved wanted to reestablish the West Roman Empire and saw a chance to do so.

First of all, the West Roman Empire did not suddenly collapse. People did not wake up some morning in 476 and see that everything had changed. For a long time after any date modern historians assign for the fall of Rome, there was the lingering idea in the West that the people who lived there were in the Roman Empire, even thought the government was not working.

The West Roman Empire had decayed badly during the 3rd through 6th centuries. During that time, there were often a number of emperors at the same time, sometimes acknowledging the status of each other, and sometimes not. People got used to the idea that the Roman Empire was not defined by its emperors.

When Germanic kingdoms were set up in the West Roman Empire, they were regarded as parts of the Empire, though not under really good control. Roman law was still in use in many places until well into the 8th century, and there was no general consensus that the Roman Empire was a thing of the past. In fact, the East Roman Empire was still one of the most important countries in Europe at the time of Charlemagne; today we call it the Byzantine Empire.

The Franks had been building up their kingdom for a long time, and it had come to dominate the West even more than the Byzantine Empire dominated the East. Charlemagne had become King of the Franks, but was also, separately King of the Lombards. In fact, his power was arguably as great as that of any of the Roman Emperors of the West.

There were a number of other things going on that argued in favor of reestablishing the West Roman Empire. Christians wanted a strong country to counter the Muslims in Spain. Pope Leo III had good reason to want Charlemagne to have as much power as possible, and in fact owed Charlemagne his life. There were people in the West to believed the empire should not be headed by a woman, and the East Roman Empire was, at that time, headed by Empress Irene; this might have influenced people at the time.

So, Pope Leo III, possibly with the help of Charlemagne, and almost certainly with the help of some of Charlemagne's supporters, crowned him Emperor of the Roman Empire.