Let's take as an $X$ set $R$ real line.
We will say $ x_1,x_2 \in R $ are in $R$ relation if $x_1-x_2$ is integer. What are equivalent classes?
Notes say that every equivalent class contains exactly one number from range $[0,1)$.And inverse is also true, that every number from $[0,1)$ is in one of equivalent classes.
Can you explain two parts? How every number from range $[0,1)$ is in one of the classes if class contains integers?
your question isn't super clear. But I'll try to answer the two things I see with question marks.
one, what are the equivalence classes. lets take an arbitrary number, $x\in\mathbb{R}$, and try to understand what is in its class. by definition $\lbrack x \rbrack=\{y\in \mathbb{R}|x-y\in \mathbb{Z}\}$. maybe we can rewrite that as $\{y\in \mathbb{R}|x+n=y, n\in \mathbb{Z}\}$. in other words, the class of $x$ is the set of all real numbers which are an integer away from $x$.
for example, lets take $x=4$ the class of this is all the numbers which we can get to by adding integers, or $4+1=5$, so 5 is in the class, $4+2=6$ so 6 is in the class, $4+(-1)=3$ so three is in the class. Thus we get to $\lbrack 4 \rbrack =\{...,-1,0,1,2,3,4,...\}$ or just $\mathbb{Z}$. Therefore the class of any integer is just all the integers. Since we need to represent each class by one particular representative, lets just call this one $\lbrack 0 \rbrack\sim\lbrack 4 \rbrack$
Lets look at a different class, $\lbrack \pi \ \rbrack$ written as a set this is just $\{...,\pi-2, \pi-1,\pi-0, \pi+1,...\}=\lbrack \pi \rbrack$. Now we need a representative for this class, we could keep using $\pi$, but $\pi - 3 \simeq .141592\in \lbrack 0, 1)$, so we are free to use $\lbrack \pi-3 \rbrack$.
This leads nicely to your second question, how is it that every number in $\lbrack 0,1)$ is in one of the classes?
Lets take an arbitrary real number, expressed as its decimal expansion $X=x.x_0x_1x_2x_3x_4x_5...$ where each $x_n\in \{0,1,2,3,4,5,6,7,8,9\}$ and $X\in \mathbb{Z}$. Now by our reformulation of the equivalence classes definition above, every time we add an integer to $X$ we get another representative of that class. By construction the 1's digit of $x$ is an integer, thus $X+(-x)=.x_0x_1x_2x_3...$ which is in $\lbrack 0,1)$. Therefore every real number has a representative in $\lbrack 0,1)$.
One question you might ask is what about infinity repeating $9$'s, does this still work? (yes/why?)
If you are thinking of this in topology, you can "visualize"the quotient space as sort of cutting up the real axis along the integers, and then "stacking" each little length one interval, left end point on top of left endpoint, and then gluing or smashing them all down to a single little length one interval. In this case, whats to quotient topology?