Why did the Roman Empire collapse? Didn't the Roman Empire have strong armies?

 The Roman Empire lasted for either 503 years (from 27 BC, the start of the reign of the first Emperor Augustus, to the abdication of the last Western Roman Emperor Romulus Augustulus in 476), or 1480 years (from Augustus to the fall of Constantinople in 1453). Either time length is a remarkable achievement for a polyglot empire spanning a continent. But all things eventually come to an end, even empires.

In the last couple of centuries of the (Western) Roman Empire, the army had become a massive cost to the empire. Worse, it was becoming obsolete. During that period, Rome faced to two major opponents: the Persian Empire to the east and German tribal confederations to the north. A good number of the Roman legions were tied to garrisons on the frontier. These soldiers were allowed to settle the land and become farmers. Consequently they were not motivated to protect other parts of the Empire. The Germans and Persians were developing more effective calvary tactics against the Romans.

Much of the 3rd century was a period of military anarchy. In 235, Emperor Severus Alexander was assassinated by his own troops. From then to the end of the century there were at least 26 claimants to the title of Emperor, most of them generals, and most meeting swift violent ends. Emperor Valerian was defeated by the Persians in 260 and died in Persian captivity. Rome was also sapped by a number plagues, including the earlier Antonine Plague (165–180) and the Plague of Cyprian (249–262).

In response to these problems, Emperor Diocletian (reigning 284–305) created new mobile armies and split the Empire into two parts, each with an emperor and a junior emperor with the title of Caesar. These reforms gave the (complete) Empire another century or so of life.

[Source: Colin McEvedy, The Penguin Atlas of Ancient History]

In the AD 300 map, you see in the upper right hand corner a group labeled Huns. These Asiatic nomads came crashing out of Asia in 372, conquering many of the German tribes and putting pressure on the remainder, many of whom applied to settle in the Roman Empire as so-called foederati. The Roman army increasingly had to rely on German mercenaries and commanders. The loyalty of these mercenaries was unreliable, and they frequently revolted against their Roman masters. One capable military commander who remained loyal to Rome was Stilicho, the son of a Vandal cavalry officer and a Roman woman. Stilicho defeated the Visigoth King Alaric in 402, and the Gothic King Radagaisus in 406. However, ungrateful Romans had Stilicho executed in 408, leaving Rome open to invasion by the German tribes. Alaric sacked Rome in 410 after a long siege.

German tribes began carving out pieces of the Western Roman Empire as independent kingdoms: the Vandals in North Africa, the Visigoths in southwestern Gaul (France) and eastern Hispania (Spain), and the Seubi in western Hispania. Attila the Hun attacked the West in 451, and was stopped by a coalition of German and Roman armies in Gaul. Attila died the next year. Vandals sacked Rome in 455. The German Odoacer finally extinguished the Western Roman Empire, deposing Romulus Augustulus in 476, only to be deposed and killed himself by the Ostrogoth Theodoric in 493.