The relation $\displaystyle \equiv_5$ is defined as $\displaystyle \forall x,y \in \mathbb{S}_7: x\equiv_5 y \Leftrightarrow x \mod 5 = y \mod 5$
To check if the relation is reflexive you have to check $\displaystyle \forall x \in \mathbb{S}_7: x \equiv_5 x$ which is true because $\displaystyle x \mod 5 = x \mod 5$.
Can you check the symmetric and transitive property now?
I have a different but equidistant way of describing that relation.
Say $\displaystyle x\mathcal{R}y$ if and only if $\displaystyle x~\&~y$ have the same remainder when divided by $\displaystyle 5$
Thus it should be clear that $\displaystyle 2\mathcal{R}7$.
The three needed properties are easily checked.
There are five equivalence classes.