I wouldn't worry too much about coming up with a physical interpretation of \[\nabla \times \nabla \times \vec{A}\]. The only reason it appears in EM theory is because the standard derivation of electromagnetic waves is to take the curl of either of the curl equations in Maxwell's equations, and then to apply vector identities and the other Maxwell equations in order to obtain wave equations for the electric and magnetic fields. There is not really any physical insight to be gained here other than the fact that Maxwell's equations and the identities of vector calculus allow for the possibility (and reality) of electromagnetic waves.
As for Schey's div, grad, curl, and all that, it is a nice, informal book to learn vector calculus. It's pretty cheap (~$30) and pretty readily available, even at chain bookstores (Barnes & Noble, Borders, etc.).