Revisiting and Editing my answer here:
The island in the North Sea on which the wizard prison is built has
never appeared on any map, wizard or Muggle. Perhaps the first
resident, the sorcerer 'Ekrizdis', who practised the worst kinds of
Dark magic, constructed a fortress and lured Muggle sailors there to
torture and murder them. After his death, the various concealment
charms placed on the island faded, and the Ministry became aware of
the mysterious site's existence.
Looking up Ekrizdis, we find that he was alone in the fortress of Azkaban, practicing the darkest of Dark Magics. He lured, tortured and killed Muggle sailors who passed the fortress. When he died, the concealment charms he placed on the fortress lifted and the Ministry were able to discover it. When the Ministry arrived, they found it populated by Dementors. So it stands to reason that the Dementors were once human, the muggle sailors that Ekrizdis tortured. Maybe they didn't serve Ekrizdis directly, but they certainly co-existed for a time.
Those who entered to investigate the island came across Dementors.
Fearing the potential reprisals of these dark entities, if anyone
tried to evict them from their home, the Ministry decided to let the
sizable colony stay, unmolested and unchecked.
The Ministry decided to let the Dementors stay at Azkaban because they didn't want to let these dark creatures loose upon the world.
When Damocles Rowle was elected Minister for Magic in 1718, he
insisted on using Azkaban instead, seeing the Dementors as an
advantage because using them as guards would save expense, time, and
trouble. This plan was eventually put into motion and despite
protests, Azkaban remained the prison of the wizarding world, mostly
because of the complete lack of breakouts and breaches of security.
Ever since, the Dementors served the Ministry of Magic as the guards
of Azkaban, as it let them feed on the emotions of the prisoners
within its walls.
The Dementors served the Ministry because the Ministry let them feed on the emotions of the prisoners.
Although a certain degree of the wizarding world could rest easy
knowing that Dementors guard Azkaban, thus making it virtually
impossible for dangerous criminal from escaping, others believe that
the Dementors are untrustworthy due to the potential of them changing
allegiance to any dark wizard who could offer them more victims than
The only reason the Dementors followed Voldemort anywhere is because Voldemort promised them more souls to harvest than the Ministry could. Their allegiance literally lies with the highest bidder.
But that isn't to say they are completely mindless. They are certainly capable of following orders and maintaining a prison. The Dementors ARE the management of Azkaban.
Answering you actual questions:
How is the prison managed? By Dementors.
Who cooks the food? The Dementors.
Who washes the blankets? The Dementors. It's not like they sit around doing nothing until somebody comes over and says "Oi. Do this." They have a level of autonomy.