The thoracic diaphragm is a dome-shaped sheet of muscle along the bottom of the rib cage. When you breathe in, it contracts, expanding the body cavity containing your lungs. The expansion draws air into them.
Hiccups are quick contractions of the diaphragm. The sharp intake of air this produces sometimes causes people to make a little noise, or hiccup, when it happens. Proposed causes for hiccups vary widely, while the real cause is still something of a mystery.
When your diaphragm contracts very quickly, it tries to expand the chest cavity very fast. The sudden lack of air pressure that results can make it feel like your chest is "going in."