For some missions, you need to collect items from your neighbors.
 
If you are wondering why the counter for collecting items is going down instead of up, you may have traded or sold items before the mission is over. Leave the items that are required for the mission in your gift box and don't sell items from your barn.
 
Once you have collected enough of the items to meet the requirement in the mission, open up the mission panel. In the mission panel, a "Use" button will appear when the required number of items have been collected. Click the "Use" button in the mission panel to complete the task.
 
If you have some of the items required left over, you safely trade them in for coins in your Gift Box.