They themselves became corrupted into monsters of the Abyss and they seek revenge against the gods for the destruction of their kingdom. The Abyss Order are the people of Khaenri'ah, a kingdom destroyed by the gods 500 years ago.
In the "We Will Be Reunited" Archon Quest, the Abyss Herald attempts to corrupt Andrius into servitude and it's revealed that they plan to use a Defiled Statue of the Seven to turn Osial into a mechanized god.
They can also influence Hilichurls to be more coordinated in their attacks one common Daily Commission quest is about you fighting an Abyss Mage who is influencing a group of Hilichurls.