I am trying to figure out what the transitive closure of this is. (Correct me if I'm wrong), but I see that it is transitive since $$x-y=c, y-z=c \implies x-z=(c+y)-(c+z)=y-z=c$$
However, I'm not sure how to check if this must also be the transitive closure, or how to get it. Any help is appreciated.
The transitive closure of $R$ must be the relation $S$ defined as $$(x,y) \in S \;\Leftrightarrow\; \exists k \in \mathbb Z\,( x - y = kc).$$
It is not entirely clear where the $R$ relation (and thus, $S$) is defined, but it should be somewhere in a set with $+$ and $-$ defined; perhaps $\mathbb R$ or $\mathbb Z$, and you should have included that information.
I assume it's in the field of the real numbers, and then, as an example, taking $c=1$, two real numbers $x$ and $y$ are related by $S$ iff $x-\lfloor x \rfloor = y - \lfloor y\rfloor$, where $\lfloor r \rfloor$ denotes the greatest integer lower than or equal $r$.