I'm curious as to how one of the largest empires in our history that a group of "barbaric" people overcame.
Weapons didn't have anything to do with it.
The full story of the fall of the Roman empire is very, very long and complex, and the historiography of the story of that fall is, if possible, even more convoluted.
For one thing you should be aware that it is, nowadays, even contested whether it is appropriate to speak of a "fall" at all, with certain historians preferring to speak of a "transformation" into a different, medieval style of life and government.
In this thread I go into considerably more detail on the theories surrounding the fall of the Roman Empire, and particularly the role of the Germanic "Barbarians" therein, as well as the different views that are currently held by historians.
As for the actual weapons: By the time the Western Roman empire fell, there was very little difference in terms of equipment between the average Germanic warrior and Roman soldier.
A very famous find at Illerup Ådal in northern Germany illustrates the kinds of weapons the Germans used in late antiquity. One interesting bit of information is that many of the swords these Germanic warriors used were, in fact, Roman: imported or looted from the Empire at an earlier date.
In the late Empire, both Germans and Romans would have used big shields, spears, long swords, and a variety of throwing weapons. The Romans would, on average, probably have had more armour, though this is hard to quantify, and not all the confederations and armies they fought would have had as much cavalry as they did, though some German peoples certainly did manage to field a strong cavalry.
The Germans certainly did not have technological superiority of any sort. Some historians, like Peter Heather, would argue that it is important that the Germanic people had less of a organisational and technological disadvantage compared to the Empire's early days. Others, like Adrian Goldsworthy, would argue that the Roman armies were still, on average, superior to Germanic forces, because the Romans were the only ones who could afford a professional standing army.
In general, I think it is a mistake to try and explain the successes and failures of ancient peoples too much in terms of technology, even when it comes to war, even when it comes to weapons and armour. We are used to thinking in terms of "weapon X beats Y." "Whatever happens, we have got, the maxim gun, and they have not." But in pre-modern times, this was not so. It's nice to have a better pointy stick than your enemy, but it won't change the face of history.
Ancient battles were won, as much as anything, by factors such as logistics, morale, leadership and luck. But even battles won or lost do not explain why peoples rose or fell, gained power or sank into obscurity. For that you need to examine entire societies, seek to understand what drove them, how they were organised, what their strengths and weaknesses were, what stresses they were under and how they reacted to them, and how they stood in the larger world around them, with its myriad of different actors and natural phenomena, all influencing how the history played out. You can't reduce it to "who had the better weapons."
Further reading: