The Nephite army that was camped around Cumeni was able to cut off supplies, and thereby keep them for themselves. That made the situation more difficult inside the city. Two things happened. The first was that the Nephite armies had more than ample provisions, and they had prisoners. The prisoners were sent to the land of Zarahemla and the excess of provisions went to Judea.
The second event occurred sometime later. The siege worked, and the lack of supplies essentially starved out the Lamanites in the city, who surrendered.