During this mission, shortly after the rotten fish start turning up, you will settle the underwater island of Sondar. Once you've established this settlement, you will be able to produce functional food. You'll also gain the ability to set up a trade route shortly afterwards. Set up a trade route to take the functional food to your island. This will keep your population fed in the interim.
I settled the Eco island initially, and as a result I got a quest from the Tycoon faction by clicking on their island's warehouse. If you bring them the goods they request (some lumber and tools, if I recall) they'll give you the "Emergency Plan" which, when put in an item slot in your island's warehouse, will keep people from becoming unsatisfied due to food. (I don't know if the Eco faction has a similar mission if you don't settle with them at the beginning of the first mission or not, but I assume they do something similar)
The fisheries on your first island will continue to produce rotten fish, however. I found that idling them caused the rotten fish to stop accumulating in my warehouse, which kept the people from getting further sick. To idle the building, first click the building, then click the percentage in the middle of the lower right hand pop-up. This will stop the building from producing goods.
You could also sell them to the Tycoon warehouse, for 72 credits per unit. As the game starts you with something like 45,000, I didn't find this to be all that useful, but your priorities may differ :)