The idea is to reduce the problem to integer arithmetic. Since $\mathbb Q(\zeta)$ is the field of fractions of $\mathbb Z[\zeta]$, we can write $$x+y\zeta = \frac{a+b\zeta}{c+d\zeta}$$
where $x,y\in\mathbb Q$ and $a,b,c,d\in \mathbb Z$, and where the numerator and the denominator on the right hand side are coprime. Here we are using the fact that $\mathbb Q(\zeta)$ has class number $1$.
In particular, $\frac mn\in\mathbb Q^\times$ (written in lowest terms) is a norm from $\mathbb Q(\zeta)^\times$ if and only if $m,n$ are both norms from $\mathbb Z[\zeta]$.
We can calculate using quadratic reciprocity that a prime $p$ is a norm from $\mathbb Z[\zeta]$ if and only if $p=3$ or $p\equiv 1 \pmod 3$. It follows that $$n = p_1^{e_1}\cdots p_k^{e_k}$$
is a norm from $\mathbb Z[\zeta]$ if and only if $e_i$ is even whenever $p_i\equiv 2\pmod 3$. (Note that all norms are positive in this case.)
We deduce that the image $N(\mathbb Q(\zeta)^\times)$ is the subgroup of $\mathbb Q^\times$ generated by $$\{p:p\equiv 1\pmod 3\}\cup\{3\}\cup\{p^2 : p\equiv 2\pmod 3\}.$$
To illustrate the difficulty of generalising this when $K$ does not have class number $1$, neither $2$ nor $3$ are norms from $\mathbb Z[\sqrt{-5}]$, but $$\frac 32 = \frac 64 =N\left(\frac{1+\sqrt{-5}}2\right) .$$
We can get around this by considering factorisation of ideals. Let $K$ be a number field, and let $x\in K^\times$. Write
$$(x) = \frac{\mathfrak a}{\mathfrak b},$$
where $\mathfrak{a,b}$ are coprime ideals of $\mathcal O_K$. Since $\frac{\mathfrak a}{\mathfrak b}$ is principal, both $\mathfrak a$ and $\mathfrak b$ lie in the same class of the ideal class group. In particular, we can find some integral ideal $\mathfrak c$ such that $\mathfrak{ac}$ and $\mathfrak {bc}$ are both principal.
Hence,
$$|N(x)| = \frac{N(\mathfrak a)}{N(\mathfrak b)}=\frac{N(\mathfrak {ac})}{N(\mathfrak {bc})}=\frac{|N(a)|}{|N(b)|},$$ for some $a,b\in\mathcal O_K$.
It follows that the image $|N(K^\times)|\subset \mathbb Q_{>0}^\times$ is generated by the elements of $\mathbb Z$ which are norms from $\mathcal O_K$. There should be some way to sort out the issue with signs.