Yes, you need to "prove something".
To show that $\sim$ is an equivalence relation, you need to prove that it is reflexive, symmetric, and transitive. That is, you need to prove that:
- For every $a\in\mathbb{Z}$, $a\sim a$ is true. Since "$x\sim y$" means "$5x\equiv 2y\pmod{3}$", then that means that you need to prove that for every $a\in\mathbb{Z}$, $5a\equiv 2a\pmod{3}$ is true.
- For every $a,b\in\mathbb{Z}$, if $a\sim b$ is true, then $b\sim a$ is true. If you are unsure what this means, then "unpack" the meaning of "$a\sim b$" and of "$b\sim a$" the way I did above.
- For every $a,b,c\in\mathbb{Z}$, if $a\sim b$ and $b\sim c$ are both true, then $a\sim c$ is true.
That will prove that $\sim$ is an equivalence relation.
Then you need to think about the equivalence classes. That is, given $a\in\mathbb{Z}$, what is $[a]=\{b\in\mathbb{Z}\mid a\sim b\}$, the set of all integers that are related to $a$? The different sets you get this way are the equivalence classes of $\mathbb{Z}/\sim$.