Whether you deliver a package, a baby, or a promise, you're bringing or carrying out something that was expected.
The Post Office and UPS deliver packages, but a doctor helps deliver a baby: in that case, the doctor is helping the baby get born safely. If you promise to get an A and then you do, you delivered on your promise. Religious people ask God to deliver them from sin. Politicians often have to deliver — meaning give — speeches. In just about every case, deliver refers to something positive that's being moved, fulfilled, rescued, or carried out.