How does coarctation of the aorta develop in the heart?

Not in the heart: Coarctation of aorta by definition does not occur in the heart. It is usually in the descending segment of the aorta and it is congenital (person is born with it). The exact location is variable, sometimes it involves the aorta proximal to the take off of the left subcalvian (artery to the left arm) and at times distal to it. It is essentially a narrowing of the lumen of the artery.

Coarctation : This develops in utero so is a congenital defect. It is a narrowing of the aorta usually just below the left subclavian artery that goes to the left arm. It can be associated with bicuspid aortic valve, vsd, and can cause hypertension. 10% of people with coarctation can have berry aneurysms of the brain arteries.

Narrowing: Coarctation generally occurs near the site where the PDA connects to the aorta. The PDA is a normal channel in fetal circulation which helps bypass the lungs (since fetuses don't breathe air). When the baby is born, the PDA constricts and closes. In some, it leads to narrowing/coarctation in the aorta as well.