Asylum in antiquity is a place where people who had done something wrong or were in danger could go to seek safety and protection. It was kind of like a big, safe house. When people went to asylum, they were under the protection of the gods, which meant that nobody could hurt them while they were there.
Back in those days, there were certain places that were considered to be asylums, such as temples or altars. People who were being chased by their enemies would go to these sacred places and hold onto the altar or temple and ask for sanctuary. As long as they were on the sacred grounds, nobody was allowed to harm them. This was seen as a very important responsibility for the people who looked after the asylum, as it was believed that the gods would punish them if they didn't take their duty seriously.
Even if someone had done something really bad, like stealing or even murder, they could go to an asylum and be safe as long as they stayed there. The idea was that once they had taken refuge in the asylum, they had made themselves subject to the judgement of the gods rather than the judgement of men. Once they had been judged, they would either be allowed to leave or punished accordingly.
Overall, asylum in antiquity was a place of safety and protection for people who had nowhere else to turn. It was seen as a sacred duty to protect those who sought asylum, and the belief in the power of the gods meant that people were often very serious about taking care of these sacred places.