Let $\text{Cl}(K)$ denote the class group of a number field $K$. If $\frak{a}$ is a non-zero fractional ideal of $K$, write $\left[\frak{a}\right]$ for its class in $\text{Cl}(K)$.
My lecture notes say that $\left[\frak{a}\right]=\left[\frak{b}\right]$ if and only if $\frak{a}=\gamma\frak{b}$ for some $\gamma\in K$.
I am struggling to see why this is true.
This is true because we are factoring out principal ideals to get to $Cl(K)$. This means that our identity element $[1]\in Cl(K)$ equals the identity $(1)$ of ideal multiplication modulo fractional principal ideals. That means we can multiply any non-zero fractional ideal $\frak{a}$ by any $(\gamma)$ where $\gamma \in K$ and obtain the same class in $Cl(K)$, since $[(\gamma)]=[1]$.
The other direction follows by noting that $[\frak{a}]=[\frak{b}]$ means that they only differ by multiplication with an prinicipal ideal $(\gamma)$.