Despite the previous answer to this post, no this was not the main reason that the Roman Empire fell. The Goths didn’t didn’t just waltz into the Empire with the sole intention of ruining everything for everyone else. In fact this was never their intention. Rome was already suffering from internal troubles by the time the Goths arrived.
In 376 the Goths arrived at the Danube river hoping to be allowed into the Empire so they could escape the Huns. Emperor Valens granted them permission because he needed more men for his war against the Sassanids, and the Goths had a large number of able bodied men with them. The problem, however, was that once the Goths crossed the river they were treated extremely poorly. They suffered from a famine, though this may not have been intentional, and the local commander Lupicinus likely charged them exorbitant prices for food. They may have also had to sell some of their own children for food, though this may have been an exaggeration.
Lupicinus then decided to move the Goths Marcianople. When they reached the city he invited Fritigern and Alavivus, the gothic leaders, to a banquet. This was a trap and he attempted to assassinate them. Fritigern survived somehow, and made a deal that he would calm down the Goths, who were now quite agitated, but instead the slaughtered the Romans at Marcianople.
After reaching a settlement with Emperor Theodosius they were allowed to settle in Thrace. Their leader Alaric was later named Magister Militum of Illyria. He rebelled sometime in 401, likely because his title was going to be taken away and this was the only way he could pay his men and keep them fed. In 408 the half-Vandal General Stilicho, Rome’s greatest General at the time, was executed on the orders of Emperor Honorius. Then the families of a large number, somewhere around 20-30,000, gothic soldiers were slaughtered by Roman citizens. These Goths then rebelled and joined Alaric.
I skipped a lot of details, but basically no the the Goths are not the sole, or even main, reason that Rome collapsed. Not was it ever their intention to bring about Rome’s collapse, they were treated like crap and decided to do something about it. The western empire had plenty of internal problems before and after Adrianople. Their economy was nothing compared to that of the East, they had a long string of usurpers, several paranoid emperors, etc. In fact the Goths helped defend the empire on several occasions. They fought against the Huns, and fought several campaigns in Spain in the service of Rome. The Western Roman Empire still would’ve fallen, maybe it would’ve taken a little longer, maybe the circumstances would’ve been different, but there wasn’t much that could’ve been done to save the West in the long run.
More can be (has been, will be) written on the subject of "The Fall of the Roman Empire". The reply in this discussion by /u/royalsanguinius looks good.
For more reading, one previous discussion of note is 'How accurate is the popular view that "uncontrolled immigration led to the fall of the Roman Empire"?' by /u/Iguana_on_a_stick and /u/shlin28 . There's also a link to followon discussion in the Sunday Digest link there. The discussion is much more than the post title. It goes into the deeper debate about what "The Fall" might mean.
The bigger section is at 'Late Antiquity/Early Medieval/"Dark Ages"'. I mention this because of the debate about falling. For example, see the first discussion, "Why does the concept of the end of the Roman Empire in 476 CE persist?" (replies mostly by bitparity ). An even more telling discussion title from elsewhere: "Did Rome Actually Fall?" . There, yodatsracist provided a lot of pointers to previous discussions along those lines.
This is not to discourage discussion. More questions, data, and debate are welcome.