$H$ is a subgroup of $G$ and $a,b$ are in $G$.
I was rather lost for a while but I think I may have actually proved it, though I am unsure… Does this make sense?
\begin{aligned} aH &= bH \\ &\implies b \in aH \\ &\implies \exists h \in H : ah = b \\ &\implies a^{-1}ah = a^{-1}b \\ &\implies h = a^{-1}b \\ &\implies hb^{-1} = a^{-1}bb^{-1} \\ &\implies hb^{-1} = a^{-1} \\ \end{aligned}
This means $a^{-1}$ is an element of $Hb^{-1}$, so they are equal.
I wasn't quite sure I could do those operations. Can anyone tell me where to learn to do the special symbols, like quantifiers and relations because it would make it simpler.