Congruence is an equivalence relation compatible with $+$ and $\times$

Reflexivity: $a-a=0$ and $m\mid 0$ for every $m$, so $a\equiv a\pmod m$.

Symmetry: if $a\equiv b\pmod m$ then $a-b=mk$ for some integer $k$, so $b-a=m(-k)$; since $-k$ is also an integer, $m\mid(b-a)$, i.e. $b\equiv a\pmod m$.

Transitivity: if $a\equiv b\pmod m$ and $b\equiv c\pmod m$, write $a-b=mk_1$ and $b-c=mk_2$. Adding these, $a-c=(a-b)+(b-c)=m(k_1+k_2)$, so $m\mid(a-c)$ and $a\equiv c\pmod m$.

Compatibility with addition: from $a-b=mk_1$ and $c-d=mk_2$, add the two equations: $(a+c)-(b+d) = m(k_1+k_2)$, which is a multiple of $m$, so $a+c\equiv b+d\pmod m$.

Compatibility with multiplication: write $ac-bd = ac-bc+bc-bd = c(a-b)+b(c-d) = c\cdot mk_1 + b\cdot mk_2 = m(ck_1+bk_2)$. This is again a multiple of $m$, so $ac\equiv bd\pmod m$. Together, (i)-(iii) make congruence an equivalence relation, and the last two steps show every ring operation on $\mathbb{Z}$ descends to a well-defined operation on residue classes $\mathbb{Z}/m\mathbb{Z}$.