Intuitive interpration of the Thom isomorphism and relative cohomology
- Sometimes these things make more intuitive sense by passing to the representing spectra, but I do not know if you are interested in that degree of generality. If you want a more "low-tech" approach, I can recommend Milnor-Stasheff's Characteristic Classes.
- By excision, that relative cohomology is the same as the disk bundle relative its boundary. In that case, we can think about it, with coefficients in $\mathbb{R}$, as having to do with forms on the disk bundle that vanish on the boundary. We are looking for a form which integrates to $1$ on each fiber. There is always a choice of such a thing when the bundle is trivial, but when we glue, we might worry about a $1$ becoming a $-1$. An orientation says we can make choices that do not cancel out. Once we have such a thing, cupping with it is an isomorphism with inverse given by "integrating it out."