In point of fact, there is no single reason why Empires fall; it would be more accurate to say that with the lapse of time, all things pass, including the great Empires of the world. Some, such as the Roman Empire, collapsed as a result of internal corruption and migration of Germanic tribes who were themselves forced into Roman territory by the Huns. There was neither external competition nor internal dissent from those within the Empire; in fact it was remarkably stable, from its founding in 45 B.C.E. until it ended in 476 A.D. The Eastern Roman Empire survived until it was overthrown by the Ottoman Turks in 1453. The Tang dynasty in China collapsed after a series of ineffective rulers led to a military coup. The Ottoman Empire as well as the Ostro-Hungarian Empire collapsed as a result of rising nationalist sentiments among subject peoples, a situation which had previously never presented itself. Jared Diamond, in Collapse, How Human Societies Choose to Fail, describes a number of environmental factors which lead to the collapse of societies as well as Empires, including but not limited to the inability to marshal sufficient resources for survival.
Bottom line, Empires run their course through history, some longer than others. People and circumstances change, and so do governments and institutions. There is no one reason which can be shown to be a common thread throughout the Empires of history.
This is a very interesting question and one that can have many possible answers. For me, there are two main reasons.
First, an empire has to try to control a large area of territory, usually made up of many different kinds of people. This is very difficult to do given that the people will generally want to get out of the empire's control. Therefore, the empire is constantly having to expend resources trying to keep its population from rebelling. This tends to end up being too much for the empire to maintain over a long period of time.
Second, empires tend to attract outside competition. Whenever one country becomes strong enough to create an empire, other countries start to worry about it. They fear that they, too, may be conquered or that they will at the very least be dominated by the empire. For this reason, they start to want to fight the empire and to create coalitions of allies to resist the empire. This, too, puts more strain on the empire's resources and helps to make it fall.