See Feynman's Lectures on Physics Vols. I and II, sections on vectors and vector calculus, for intuitive presentations. Volume II is mostly on electromagnetism.
If not for the intuition, then for the style of writing. Feynman on abstracting/extracting a vector operator from a physical example of its action on an operand in an equation:
Now we can do something that is extremely amusing and ingenious—and characteristic of the things that make mathematics beautiful. ... .
We leave the operators, as Jeans said, “hungry for something to differentiate.”