The elements of $\mathbb{R}$ are real numbers. The elements of $\mathbb{R}/\mathbb{Z}$ are sets of real numbers, pairwise disjoint, where each set contains reals that differ from each other by integer distances, and the set contains all reals that differ from the reals in the set by integer distances.
So one of the elements of $\mathbb{R}/\mathbb{Z}$ is $\{ \ldots, -2, -1, 0, 1, 2, 3, \ldots\}$ Another element of $\mathbb{R}/\mathbb{Z}$ is $\{\ldots, \pi-3, \pi-2, \pi-1, \pi, \pi+1, \pi+2,\ldots\}$ and so on. Every real number is in one and only of the elements of $\mathbb{R}/\mathbb{Z}$ (which, remember, are sets of real numbers).
(The elements of $\mathbb{R}/\mathbb{Z}$ are the equivalence classes of real numbers under the equivalence relation "$x\sim y$ if and only if $x-y\in\mathbb{Z}$").
Now, the set $\mathbb{R}/\mathbb{Z}$ can be made into a group as well; that is, we can define an "addition of classes". One way to define this "addition of classes" is to first give a "name" to each element of $\mathbb{R}/\mathbb{Z}$. Since every one of the sets contains one, and only one, real number in $[0,1)$, we will represent the set that contains the real number $r\in[0,1)$ by writing $[r]$. So the first set I wrote above is called $[0]$, the second set I wrote above is called $[\pi-3]$, and so on.
(Added. Given a real number, how can you tell what element of $\mathbb{R}/\mathbb{Z}$ it is in? Remember that $\lfloor x\rfloor$ is defined to be the largest integer $n$ such that $n\leq x$. The "fractional part of $x$" is defined to be $ x-\lfloor x\rfloor$. It is not hard to check that $x\in[x-\lfloor x\rfloor]$.)
Now that every set has a name, we define addition of classes. The way we are going to define addition (which I will write $\oplus$ to distinguish it form the addition of real numbers) is as follows: if $[r]$ and $[s]$ are two classes, then $[r]\oplus [s]$ is: $[r]\oplus [s] = \left\{\begin{array}{ll} \ [r+s] &\text{if }0\leq r+s\lt 1\\ \ [r+s-1] &\text{if }1\leq r+s. \end{array}\right.$ Because $0\leq r,s\lt 1$, then $0\leq r+s\lt 2$, so one and only one of those situations will happen, and the result will always be a "proper name" for an element of $\mathbb{R}/\mathbb{Z}$ (that is, the answer is of the form $[a]$ with $0\leq a\lt 1$).
Then one can show that this makes $\mathbb{R}/\mathbb{Z}$ into a group.
What the text you are quoting is doing is that it is writing "$r+\mathbb{Z}$" where I wrote "$[r]$" above. The reason it does this is that this is the standard notation when performing the kind of construction that is being discussed. This is covered in any book on abstract algebra that discusses groups.