By the mid-470s, the Eastern Empire was rich, but the Western Empire was poor and struggling in every capacity. The Western Empire's army was so dependent on Germanic soldiers that the Emperor was a puppet of the commander of the Germanic forces, Ricimer. Ricimer would even kill emperors who stopped acting as loyal puppets! After Ricimer died, the emperor of the Eastern Empire decided enough was enough, and he took the opportunity to restore Roman rule to the Western Empire by sending Julius Nepos to lead an army to depose both Ricimer's successor Gundobad and his puppet Glycerius.
Julius Nepos was successful, establishing himself as the new emperor of the Western Empire. He appointed two key generals who he believed were loyal to him: Orestes leading the Roman forces, and Odoacer leading the Germanic forces. Shortly after this, there was an uprising in Gaul by a group of Germanic tribes who were upset over not having enough land. Julius Nepos sent Orestes to put down the rebellion, which wound up being a fatal mistake. Orestes had a dream his entire life of restoring the Western Empire to its former glory, and he believed Julius Nepos wasn't strong enough to do so. In fact, he believed that the only person who could make Rome great again was himself. So when he arrived in Gaul, he made a deal with the rebels. He told them that if they followed him home to the capital in Ravenna and helped him overthrow Julius Nepos, he'd grant them all the Italian land they want.
And so Orestes and an army of Germanic rebels entered Ravenna and sent both Julius Nepos and his remaining general Odoacer into exile. Orestes knew that it would be a mistake to take the throne himself -- nobody would tolerate an emperor who had the last guy kicked out -- so he instead appointed a puppet he could control, his nephew Romulus Augustulus ("Augustulus" being a nickname for "little emperor", not his actual name). But then Orestes made his own fatal mistake. He turned his back on the Germanic forces who helped him defeat Julius Nepos and said "I lied". He gave them no land and told them to beat it back to Gaul.
They weren't happy at all. They were so pissed in fact that they tracked down Odoacer in his exile and begged him to lead them to overthrow Orestes and his puppet. Odoacer agreed. In the civil war that resulted, Orestes and his brother were both captured and executed, Ravenna was sacked, and Romulus Augustulus sent into exile. Odoacer was nice enough to actually give the kid a generous stipend and sent him to live with the rest of his family, since it wasn't really his fault Orestes put him on the throne. Tired of the cycle of conflicts over the emperorship, Odoacer declared himself King of Italy and sent all the imperial regalia to Constantinople along with a note saying that an emperor in the west is no longer needed. And the Germanic refugees got all the land they could ever want.
And that is how the Western Roman Empire ceased to exist. Breaking your promises to your refugees is a good way to get overthrown.
(edited to fix a typo; thank you jacquesm for pointing it out)