Food in My Tribe can be gathered by three different methods; fishing, farming, and star/moon dust.
Set a tribeperson's priority to fishing. Compared to the other two methods of food gathering, fishing is fairly slow, but it is the only method available at the start of the game. When a tribesperson is manually dropped on a spot in the ocean where fish can be seen jumping, fish gathering is faster, until the jumping fish no longer show up.
Once the third level of agriculture is researched in the science menu, farm plots can be placed on the island. Any tribesperson with the agriculture priority checkbox marked off will automatically sow seeds in an empty farm plot and harvest crops when they are ready, assuming there is room in the food storage. This method is far more effective than fishing.
Star/Moon Dust Edit
This method cannot be maintained for any length of time, and should only be used when in a bind. Simply drop either stardust or moondust onto the food tarp or completed food storage building to add food to the tribe's food storage. Moondust gives more food than stardust.