By the third century, the Roman Empire had reached its zenith. There was also the peculiar nature of the army in that the loyalty of individual troops was to their commander, not to the Emperor. Several commanders were proclaimed emperor by their troops often to the exclusion of other claimants who had their own troops to back them up. The army itself was often divided and civil war was often the result. Gothic tribesmen were often hired as mercenaries to fight in these civil wars against other Roman troops who often had mercenaries of their own.
An additional factor was the sheer size of the empire. It's borders were so broad that they were difficult to maintain. When the Goths crossed over the Danube, it seemed easier to retain them in the army than to police them. Ultimately, Germanic mercenaries were the order of the day. Odoacher, the Gothic chieftain who deposed the last Roman Emperor (Romulus II Augustulus) was actually fighting as a mercenary for the Romans and having deposed Romulus forwarded the royal scepter and purple robe to Constantinople with word to the Emperor there that he was now the sole Roman Emperor. An excellent discussion of this topic can be found in How Rome Fell, Death of a Super Power by Adrian Goldsworthy.
The basic reason for this is that there were no longer enough Roman citizens who wanted to and were fit to be good soldiers. The foreign mercenaries were plentiful and were highly motivated by the offer of Roman citizenship in exchange for their service. Because there were plenty of them and so few Romans who were willing, the Romans started to hire mercenaries.