= Solution
<Maschke's theorem> says that every invariant subspace of a finite-dimensional complex representation of a finite group has an invariant complement. Average any projection over the group to make it equivariant. Induction on dimension therefore decomposes every representation into irreducibles.
The <character of a representation> is $\chi_V(g)=\operatorname{tr}(g|_V)$. For a $G$-set $X$, the permutation representation $\mathbb CX$ has basis $X$, and its character is the number of fixed points:
$$
\chi_{\mathbb CX}(g)=|X^g|.
$$
For the regular action,
$$
\chi_{\mathbb CG}(1)=|G|,
\qquad \chi_{\mathbb CG}(g)=0\quad(g\ne1).
$$
The multiplicity of an irreducible $V_i$ is the character inner product
$$
\langle\chi_{\rm reg},\chi_i\rangle
=\chi_i(1)=\dim V_i,
$$
so
$$
\boxed{\mathbb CG\cong\bigoplus_i(\dim V_i)V_i}.
$$
If $V\cong\bigoplus_i n_iV_i$, <Schur lemma> gives
$$
\boxed{\dim\operatorname{Hom}_G(V,V)=\sum_i n_i^2}.
$$
For $V=\mathbb CG$, this is $\sum_i(\dim V_i)^2=|G|$.
Now suppose $\chi(g)=0$ off the identity. The multiplicity of $V_i$ in $V$ is
$$
\langle\chi,\chi_i\rangle
=\frac{\dim V}{|G|}\dim V_i.
$$
Thus $\dim V/|G|$ is a nonnegative integer and $V$ is that many copies of the regular representation. Finally,
$$
\chi_{W\otimes\mathbb CG}(g)
=\chi_W(g)\chi_{\rm reg}(g)
=(\dim W)\chi_{\rm reg}(g),
$$
so
$$
\boxed{W\otimes\mathbb CG\cong(\dim W)\mathbb CG}.
$$
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