Here is my proof so far:
(=>) Let $f: X \to Y$. If $f$ is injective, then $f(a) = f(b)$ implies $a = b$. So for $f(a) = f(b), a = b$. thus for the equivalence class $[a]$, $[a] = \lbrace b \in X \mid f(b) = f(a)\rbrace$. since $b = a$, $a$ is the unique element in $[a]$.
(<=) Let each $\sim$ equivalence class have a unique element. Let $a$ be in $[a]$ be a unique element. Then $[a] = \lbrace x \in X \mid f(x) = f(a)\rbrace$. So for each unique element $a$, $f(x) = f(a)$. Since $a$ is a unique element in $a$, then $x = a$.
I'm stuck on the second part of the proof (<=). I don't think what I currently have for to get from f(x) = f(a) implies x = a makes sense. Any help would be greatly appreciated.