quantum mechanics - Seeking a quality plain-language description of the Wigner-Eckart theorem

I'm a third year physics undergrad with a very cursory knowledge of quantum mechanics and the formalism involved. For instance, I understand roughly how tensors work and what it means for a tensor to be irreducible, though it would take me a lot of work to apply this knowledge to a problem/extend it past what I've already seen.

As part of a project, I'm studying atomic nuclei in electric and magnetic fields. I'm trying to understand the energy of a nuclear quadrupole's interaction with an electric field gradient. The equation for this is

$E_Q = \sum_{\alpha,\beta} V_{\alpha\beta} Q_{\alpha\beta}$

where $\alpha$ and $\beta$ each iterate over $x, y, z$.

$Q$, the electric quadrupole moment, is given by

$Q_{\alpha\beta} = [\frac{3}{2}(I_{\alpha} I_{\beta} + I_{\beta} I_{\alpha}) - \delta_{\alpha\beta}I^2] * constant$

(taken from this powerpoint.) An electric quadrupole moment should have nothing to do with nuclear spin... or so I thought, until I ran across the idea of "spin coordinates" and the Wigner-Eckart theorem. This is roughly all I know about the theorem -- that it exists and that it can somehow convert between Cartesian and spin coordinates in quantum systems -- and I'd like to understand it better.

THE SHORT VERSION: I do not need a detailed mathematical understanding of the Wigner-Eckart theorem, but I'm very curious as to the general idea of it. Can anyone think of a plain-English (or rather, minimal-math) explanation of the theorem that would make sense to a beginning quantum physics student?