In a word, money.
The last-known gladiator fight took place in 404 AD. Contemporary Christian historian Theodoret wrote in his Ecclesiastical History: A History of the Church in Five Books of Saint Telemachus, a Christian monk who rushed onto the field during the match to try and stop it and was killed. The exact details of how he was killed and what he said vary from account to account, but in any case Theodoret claims that the Christian emperor Honorius was so disgusted at the death of the monk that he outlawed gladiatorial battles in the Colosseum after that.
In 410 AD, the city of Rome was sacked for the first time in 800 years, heralding the beginning of a long decline for the city in both wealth and population. Although there isn't too much writing about what was going on with the Colosseum that I know about, The outlawing of gladiator events may well have stuck for economic reasons: maintaining an expensive arena and a professional sports league is an easy thing to cut when your treasury is empty and your city is ruined.
The Colosseum was always host to events besides gladiatoral fights, of course, notably animal fights. The last of those documented was in 523 AD, a full half-century after the fall of the Western Roman Empire. Specifically, it was an animal hunt to commemorate the ascension of one Anicius Maximus to the consulship. Maximus' lavish animal hunt got him in some trouble with Rome's de facto ruler, Theoderic the Great, who chastised Anicius for spending a huge amount of public money on something so wasteful.
Theoderic's death in 526 signaled the beginnings of harsh times for the Kingdom of Italy, climaxing in the 535 Gothic War: Emperor Justinian's invasion of Italy in an attempt to bring Rome under Roman control once more. The nearly 20-year campaign was ruinous to the entire region but especially the city of Rome, which was besieged no less than three times and ultimately sacked (Richard Krautheimer writes that the city's population fell to just 30,000 after the sack, compared to an estimated 800,000 a century prior). The war's end in 554 signaled centuries of decay and turmoil to Italy, depriving its once-cosmopolitan cities of their peoples and wealth.
The city's population had been drastically reduced, extensive damage was done to the public works from the fighting, and Justinian's attentions were called to more pressing concerns than restoring Rome to its former glory. There was no time, money, or interest in reviving the Colosseum. By the end of the 6th century the Colosseum's grounds was host to a shrine and a cemetery.