= Solution
The point $1/N+\Lambda_\tau$ has exact order $N$, and changing $\tau$ by $\Gamma_1(N)$ changes the pair only by a complex scaling, so the stated map is well-defined.
Conversely, scale a lattice to write it as $\mathbb Z\tau\oplus\mathbb Z$. A point of exact order $N$ is represented by $(r\tau+s)/N$, where $(r,s)$ is primitive modulo $N$. The group $SL_2(\mathbb Z)$ acts transitively on primitive vectors modulo $N$, so a basis change carries this point to $1/N$. This proves surjectivity. Two resulting normalized pairs are similar precisely when their basis-change matrix fixes $(0,1)$ modulo $N$, namely when it lies in $\Gamma_1(N)$. This proves injectivity and the <Gamma 1 level structure on a complex lattice> bijection
$$
\Gamma_1(N)\backslash\mathfrak h\cong\mathbb C^\times\backslash\mathcal L(N).
$$
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