As explained in my comments, it is the same to treat the question where the notation $L^{\ast}$ is defined with $V^{\ast}$ taken to be the $\mathbf{C}$-dual (rather than the conjugate dual), so we do that. That is, $L^{\ast}$ now consists of the $\mathbf{C}$-linear (rather than conjugate-linear) forms on $V$ whose imaginary part is $\mathbf{Z}$-valued on $L$. Note also that $\mathbf{R} \otimes_{\mathbf{Z}} L$ coincides with the underlying $\mathbf{R}$-vector space $V_{\mathbf{R}}$ of $V$. We let $V_{\mathbf{R}}^{\ast}$ denote the $\mathbf{R}$-dual of $V_{\mathbf{R}}$ (not to be confused with the underlying $\mathbf{R}$-vector space of the $\mathbf{C}$-dual $V^{\ast}$!).

The answer below is rather longer than the actual content, since we wish to explain why the relevant formulas don't just come out of thin air. If you don't care about motivation, just go immediately to the final paragraph for a quick self-contained proof.

Write $\overline{V}$ to denote the conjugate-dual space (i.e., $\mathbf{C} \otimes_{\sigma, \mathbf{C}} V$ where $\sigma$ denotes complex conjugation), so for $v \in V$ we have the associated vector $\overline{v} := 1 \otimes v \in \overline{V}$. Thus, as $\mathbf{C}$-vector spaces we have naturally
$$\mathbf{C} \otimes_{\mathbf{Z}} L = \mathbf{C} \otimes_{\mathbf{R}} V_{\mathbf{R}} \simeq V \oplus \overline{V}$$
where the isomorphism is defined by $1 \otimes v \mapsto (v, \overline{v})$. The inverse isomorphism is clearly
$$(v, \overline{w}) \mapsto 1 \otimes \frac{v+w}{2} + i \otimes \frac{v-w}{2i}.$$
Now the idea is to $\mathbf{C}$-dualize both sides, exploiting that the $\mathbf{C}$-dual of $\mathbf{C} \otimes V_{\mathbf{R}}$ is naturally identified with $\mathbf{C} \otimes V^{\ast}_{\mathbf{R}}$, and that
$V_{\mathbf{R}}^{\ast}$ is $\mathbf{R}$-spanned by the $\mathbf{Z}$-dual of $L$, with this $\mathbf{Z}$-dual identified as the $\mathbf{R}$-linear functionals on $V_{\mathbf{R}}$ that are $\mathbf{Z}$-valued on $L$.
We'll thereby eventually find that up to a harmless factor of 2, this identifies the $\mathbf{Z}$-dual of $iL$ with $L^{\ast}$ **as defined in the original problem** (thereby "explaining" the notation) and from that we'll get the entire result (that $L^{\ast}$ as originally defined is both discrete in $V^{\ast}$ and spans it over $\mathbf{R}$) all at once.

To actually do the work, note that the restriction of the inverse isomorphism to the first direct summand is a $\mathbf{C}$-linear inclusion $V \hookrightarrow \mathbf{C} \otimes_{\mathbf{R}} V_{\mathbf{R}}$ given by $v \mapsto 1 \otimes (v/2) + i \otimes (v/2i)$.
Thus, passing to $\mathbf{C}$-duals gives a
$\mathbf{C}$-linear quotient map
$\mathbf{C} \otimes_{\mathbf{R}} V_{\mathbf{R}}^{\ast} \twoheadrightarrow V^{\ast}$ defined by $$1 \otimes \lambda \mapsto (v \mapsto \lambda(v/2) + i \lambda(v/2i) = \lambda(v/2) - i \lambda(iv/2)).$$
The restriction to the $\mathbf{R}$-subspace $V_{\mathbf{R}}^{\ast}$ is an $\mathbf{R}$-linear map $V_{\mathbf{R}}^{\ast} \rightarrow V^{\ast}$ that is clearly injective and hence an isomorphism for $\mathbf{R}$-dimension reasons (or by bare hands: equivariance with $i$-scaling shows that $\mathbf{C}$-linear forms on $V$ can be uniquely written as $v \mapsto x(v) - ix(iv)$ for $\mathbf{R}$-linear forms $x$ on $V_{\mathbf{R}}$).

The above is really just natural context for something we could have written in a single line: $V_{\mathbf{R}}^{\ast} \simeq V^{\ast}$ via the formula $\lambda \mapsto (v \mapsto \lambda(v) - i\lambda(iv))$.
Thus, for a lattice $L$ in $V$, the $\mathbf{Z}$-dual to $L$ as a *lattice* in the $\mathbf{R}$-dual $V_{\mathbf{R}}^{\ast}$ of $V_{\mathbf{R}} = \mathbf{R} \otimes_{\mathbf{Z}} L$ is carried over to the set of $\mathbf{C}$-linear forms $f$ on $V$ whose real part is $\mathbf{Z}$-valued on $L$, or **equivalently** (!!) whose imaginary part is $\mathbf{Z}$-valued on $iL$. In particular, we conclude that for *any* lattice $\Lambda$ in $V$, the set of $f \in V^{\ast}$ satisfying ${\rm{Im}}(f(i\Lambda)) \subset \mathbf{Z}$ is a **lattice**. Now take $\Lambda$ to be $iL$ (so $i\Lambda = L$) to conclude.