Harish-Chandra isomorphism is defined for semisimple algebras.
Let me try to give informal explanation for classical simple Lie algebras, take gl_n as main example.

We are interested in the center of U(gl_n) - the same as invariants of gl in that space. First let us look on S(gl_n) - symmetric algebra of gl_n - as a vector space and moreover as module of g, it is isomorphic to U(gl_n).
So you need to describe the invariants of S(gl).
Same as invariants of group GL on the space of all matrices where GL acts by conjugation

Actually **everybody knows the answer - invariants are coefficients of the characteristic polynomial** !
Coefficients of the char.pol are symmetric functions of eigenvalues - so you get
the S(gl)^GL = Sym(h^*)^W.
(As for me this simple fact is main "motivation" for HC-isomorphism).

**Now, what is NON-obvious ?** We need to pass from S(gl) to U(gl) and this passing is actually related to twisting the action Weyl group by shift on rho/2.

I do not think there is simple explanation of this, but there is some general context which explains at least why ZU(g) is isomorphic as commutative algebra to S(g)^g.
Moreover for any Lie algebra "g" not just semisimple one.

This is what is called Duflo isomorphism. Moreover it was generalized by M. Kontsevich to quantization of arbitrary Poisson manifold.

So roughly speaking the "general context" is the following - "quantization is GOOD" i.e. it many algebraic structures which we can see on the classical level of Poisson algebras can be transfered to quantum level isomorphically. This is formalized by "formality theorems" like Kontsevich one, and later ones.

**Now what about why the Weyl group becomes shifted ?**
I do not know the answer - it is clear how one can guess this - just take Casimir for sl(2) and you will see it. But conceptual meaning is not clear for me.

If it is clear for someone - then please would you be so kind to comment what happens in the case of affine gl ?
See my question: Harish-Chandra isomorphism for loop algebras. Is the image invariant with respect to affine Weyl group ? Is there relation - Miuru transformation - affine Weyl group ?

Similar reasons works for other classical semisimple Lie algebras, and may be not only for classical.

PS

There is generalization of Harish-Chandra isomorphism
http://arxiv.org/abs/0912.1100
A generalized Harish-Chandra isomorphism
Sergey Khoroshkin, Maxim Nazarov, Ernest Vinberg
(to confess S. Khoroshkin was my supervisor).