This is certainly not an overview of intersection theory as a whole, but for its classical roots I highly recommend:

Steven L. Kleiman, Problem 15: rigorous foundation of Schubert's enumerative calculus. Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974), pp. 445–482. Proc. Sympos. Pure Math., Vol. XXVIII, Amer. Math. Soc., Providence, R. I., 1976.

I found this article to be both completely lucid and completely fascinating -- and I am someone who, in general, has no great interest in intersection theory or (especially) Schubert calculus.