Why is it harder for me to breathe after a big meal?

Pressure on the lung. Eating adds volume into a confined space which is the abdomen. As a result, there is increased pressure in the abdomen which is transmitted to the lungs. The bigger the meal, the higher the pressure. The shortness of breath is more likely to happen if the lungs are already compromised. Talk to your doctor about this.
Diaphragm function. Our stomach lies just below the diaphragm which does 75% of our work of breathing. A large meal creates resistance to the lowering of the diaphragm during inhalation which we sense as difficulty breathing. In some chronic lung diseases this can cause significant symptoms which contribute to inappropriate weight loss. Frequent small meals and not overeating are good strategies to help.
Pushing on lung. When you eat a large meal, the stomach expands. The stomach is right underneath the lung on the left side, and so when it pushes on the lung, it makes it more difficult to breathe. It can get particularly bad if you already have compromised lung function. Once the food moves out of the stomach, typically the breathing will get better.