The double hyphen -- represents a dash [1.][2.]. Words between dashes give information about the sentence in which it is embedded. For the moment, let us ignore that information and focus on the rest:
The man goes through airport screening and manages to persuade security that his biological samples are harmless.
It should be clear now that the word harmless refers to the biological samples. Dangerous biological materials are usually referred to as biological hazards because they are "biological substances that pose a threat to the health of living organisms, primarily that of humans". So, the security at the airport believed that his biological samples were biohazards and are not allowed on airplanes. However, he was able to convince them that the materials were safe.
Now we return to the dashed part:
one for each of the many cities on his itinerary
the places you go to or plan to go to on a journey
So, the man took a sample at each city that he visited, every city on his list (itinerary). So all together, he took one sample from each city on his list, and there were many cities on that list.