Ah, this question has beset historians for centuries! Theories about what brought the Fall of the Roman Empire include excessive military spending, lead poisoning, and just about everything in between. The present consensus, based upon historical and archaeological evidence, is that there was no singular cause of the Fall, but a number of factors which served to compound one another.
Perhaps the biggest reason contributing to the Fall of Rome was how widespread the Empire grew to be. At its fullest extent, the Roman Empire included the coast of North Africa, all of Southern and Western Europe, and even territory in the Arabian Peninsula. In all of these territories, soldiers and officials were sent from the central Roman territory to act as overseers and enforcers of the Roman law. Over such a wide expense of territory, it was incredibly difficult to govern such various locations with differing troubles like crop failure, crime, or invaders from outside the Empire. Imagine trying to hop on one foot, tap dance with the other, use one and to eat spaghetti, and the other to juggle- it's just too much going on out of concert! In short, Rome could not effectively manage the great expanses of territory they had acquired so quickly, and the empire crumbled under poor administration.
In addition to the "too much to govern, not enough government" problem, the Roman Empire had to deal with repeated invasions by Germanic tribes from the North. With inadequate numbers of Roman soldiers to fight or officials to negotiate, territory in the North and West of the Empire quickly fell to the Goths, the Huns, the Alans, and others. Some of these tribes were so successful that they made it all the way to Italy, and in 476, the Germanic king Odoacer deposed the last Emperor of Rome. Though Rome had been weakening for some time, the overthrow of Emperor Romulus meant that the West was irreparably torn apart.