Rust is iron oxide, a molecule consisting of iron and oxygen. The mass of the iron in the rust molecules comes from the original nail, but the mass of the oxygen has come from the air.
When rust forms on iron, the mass of the iron object is increased by the mass of the oxygen that has combined with some of the iron.
Rusty iron weighs more because the iron reacts with oxygen in the air to form oxides which are solids which appear to grow on the iron as more oxygen comes along.
