Biyou literally had no idea the reason why they were laughing. He really thought that Rex said some joke. Well, this laugh would have cost him a lot. After all, losing Rex's trust and friendship was probably one of the worst things a person could lose.
Regardless, Rex didn't care about Biyou right now. He was infuriated by the checkers and their talks.
Of course, one of the reasons why they were laughing was because Rex didn't have any crest. The back of his palm was completely blank.
One should have a crest on it, but Rex had none. This was more than enough for them to think that the rumors were true.
Also, they knew that he was Rex right away after not finding any crests.
Well, commoners usually don't have crest either and Rex was wearing a dress that matched the commoners, but his face and talk about passing the exam was more than enough to make them believe that he indeed was Rex!