Americans saw an increase in manufacturing during the War of 1812 because the war cut them off from their previous supply of imported manufactured goods. With imports cut off, the Americans had to make their own manufactured goods. This led to an increase in American manufacturing.
During colonial times, the American colonies were not really allowed to manufacture many things. The British government did not let them do this because it wanted to be able to export manufactured goods to the colonies. By preventing the colonies from manufacturing, they protected their own manufacturers.
After the American Revolution, Americans did start trying to manufacture more. However, they still lagged behind the British very badly. The British had a tremendous head start in manufacturing and the US could not really compete. Therefore, American consumers still continued to buy imported goods from Britain.
When the War of 1812 broke out, America was cut off from these British goods. People still needed the goods but could not get them from Britain as easily. This meant that American consumers had to buy goods from American manufacturers. This is why Americans saw an increase in manufacturing during this war.