The Germans and the Goths didn't took Rome overnight. Roman decline took many centuries and you can see its premises already during Caesar's time.
On immigration
The Gauls were conquered 52BC, but it created a permanent threat to Rome: the Germans. You could not incorporate the Gauls without defending them from the Germans.
So they created the concept of "federate peoples", granting lands to Germans, and incorporating them in the army. But the outsider Germans and later Goths had a very different resource equilibrium, based on war and loot.
On the religious point of view
The Romans were a very superstitious people, with many augurs and sacrifices at about every occasion.
Then, the Heebs settled in Rome (as they did in Greece or in Egypt) where they would follow the Exodus book: destroy the others' "idols" but remain a believer in yahweh. So it's a tale of two superstitions. As a result of their influence many were attracted to the Jewish faith and synagogues, with the tithe being sent to Jerusalem. Jews became rich and politically active. Nero's wife was a jewess (of faith, if not of race) called Poppaea. She was filthy rich and excessive (the picture talks of baths of milk). After their deaths, the Roman army led by Vespasian reacted against the Judeans in 70AD, but the seeds were planted.
Also note that many godless philosophies appeared in Rome, such as Seneca and the stoicians, or Epicurus. They were looking for some individual wisdom at a time of circus games, prostitution and religious doubt.
Also, during the Roman Republic and Empire, there were always pressures to grant Roman Citizenship from plebeians or non-Romans who said "it was mean not to". There was also a conservative party who managed to moderate these claims, but they never but recanted these extensions.
So it was handed out to an increasing number of people:
Year | citizenship granted to all free men of…
340 BC: the Latium
[ great Roman expansion in the late 2nd century BC ]
[ census of 120BC: 400000 citizens. Rome was ruling over the whole Mediterannean (Greece was added in 146 BC) ]
[ 95-90 BC: tensions with Italian allies ('allies') over citizenship lead to the Social War ]
89 BC: all Italian free men
[ census of 70BC: 900000 Roman citizens ]
[ 52BC: conquest of the Gauls ]
49 BC (Caesar): Cisalpine Gauls
48 AD (Claudius): Gauls
212 AD (Caracalla Edict): the whole Empire, except the dediticii (which were, in this context, son of invaders . . .)
From the 3rd century BC, countless accounts and testimonies speak of a demographical decline, because of anarchy, invasions, overtaxation, economic downturn.
Citizenship was no longer extended. Since the rule of law disappeared, there was no point to have more rights.
And YOU, if you didn't have the nationality you have now, would you want to have it? would you want to be the national of any Western country?