Once your arugula has entered its flowering and seeding season (normally when the weather is hotter), seed pods will form. Leave them undisturbed until, like the arugula flowers, they have dried out and turned yellow.
Simply shake the seed pod to test if it is ready to be harvested. If you can hear the rattling of the seeds inside the pod then it's time to cut the stalk below the pod and get ready to harvest the seeds.
Since each stalk will have produced hundreds of seeds, the quickest and easiest way to separate the plant debris from the seeds is to hold the stalks above a colander with a bowl underneath and rub the stalk. This should open the seed pods which will fall through the colander and into the bowl, while keeping all the debris in the colander.
This way you can keep your arugula seeds in a dark, cool place, ready for the next fall or spring to keep your arugula plant's lifecycle going.