I'm reading section 8 Differentials of chapter 2 in Hartshorne. It's is extremely hard to me to understand the nature of the definitions: module of relative differential forms - sheaf of relative differential - tangent sheaf.

It's said that "the sheaf of differential forms is essentially the same as the dual of tangent bundle defined in differential geometry".

I want to understand very explicitly how these constructions come from Diff.Geometry?