Gauls, Celts and other groups invaded Rome for many different reasons, including pressure from the distant Huns from the East. Rome had changed since its early years of glory, progressing from a republic governed by citizen-soldiers into an elitist bureaucracy governing a mostly idle citizenry. It had grown soft and weak, according to the website for the Department of Economics of San Jose State University.
By the late 5th century A.D., the Roman army was staffed by officers from the aristocratic families of Rome, but with increasing numbers of auxiliary soldiers from the ranks of the non-citizen Gauls and Germans, who could thus earn citizenship. This was a necessary evil as Rome's wealth made it easy for the privileged citizens to avoid military service, and the Senate had opted to avoid public unrest with "bread and circuses" - free bread rations from public ovens and free shows in arenas like the Coliseum.
When pressure from the north and east built on the outer provinces, it was easier to allow more barbarians from the East to enter and settle on lands previously held by the Gauls and Germans than to fight them off. The Gauls allowed no say in the running of the Empire, becoming increasingly dissatisfied and less likely to enter Roman military service than to stay home and protect their own people.
At last Alaric, a German Visigoth, led an army of barbarians against Rome, demanding land or tribute. When the virtually defenseless city refused, Alaric and his men went in and took what they wanted. Realizing Rome was no longer the power it once was, other barbarian hordes soon followed. Ultimately, Rome was forced to seek protection from the Gauls, who they had formerly ruled.