The perennial question of ancient history. Many historians have chimed in on this one. The empire was, at its peak, immensely strong. At the center of the empire was the Mediterranean Sea, offering water transport to and from all points. The empire controlled the entire shoreline of the Mediterranean, which for the first time in history, put down the pirates. Pirates are not suppressed in blue water encounters with naval vessels, they are suppressed by taking over their home ports. The empire was the first to profit from widespread sea trading. Water transport was so cheap that the city of Rome fed its people on grain imported from far away Egypt. So what laid the empire low?
Perhaps it was the lack of a succession mechanism. When the emperor died, a civil war often broke out to determine the next emperor. Nothing saps the strength of an empire like a civil war. In addition to the killing and the property damage, the survivors are likely to adopt a life long "keep your head down" attitude and contribute as little as possible to keeping things running.
Then there must have been technological diffusion. In Julius Caesar's time (first century BC) the legions were far superior to the Gauls and the Celts and everybody else. The Legions knew about marching in step, staying in ranks, obeying orders, and they had better weapons too. Caesar tells of Gauls having to drop back from the front line, lay their bent swords on the ground, and pull them straight again. The legionary gladius didn't have this problem, due to superior Roman smith craft.
By say 300 AD, this superior Roman technology must have diffused out into the wider world, and the legions found them selves fighting barbarians who were as good as they were. Some of this diffusion must have come from the Roman practice of enlisting barbarians to fill the ranks of the legions. After serving a 20 year enlistment, the retired veterans must have known everything there was to know about Roman military art. Plenty of them must have returned home to where ever and passed on what they had learned.
There are other reasons, such as failure of the Roman tax base, that have been widely discussed by historians, the notions of repeated succession crisis and technological diffusion are my own.