To "argue their case", a person would tell the another person their view on a situation. The phrase comes from a court of law when a lawyer would state all the facts and show all the evidence supporting their case, or story, to the judge. In everyday usage, a child who tracks mud into the house would argue his case to his mother saying that it was raining and he could not avoid getting muddy. Or your product is better than the other store's product and you argue that case to the customer. Or if an employee is late for work, he may argue his case to his boss. Hope this helps. Good luck! =)

It means that they gave their points of view about the topic