In modulo arithmetic, "division" means multiplying by the multiplicative inverse, e.g., $b = \frac{1}{a}$ means the value which when multiplied by $a$ gives $1$ modulo the value, e.g., $ba \equiv 1 \pmod n$. Note you may sometimes see $b = a^{-1}$ instead to avoid using explicit "division". This works, and gives a unique value, in any cases where the value you're dividing and the modulus are relatively prime.
More generally, it'll work in all cases of $\frac{c}{a} \equiv e \pmod n$ where $d = \gcd(a,n)$ and $d \mid c$ since this gcd value "cancels" in the division. Thus, the resulting equivalent modulo equation of $\frac{c'}{a'} \equiv e \pmod n$, where $c' = \frac{c}{d}$ and $a' = \frac{a}{d}$ has $\gcd(a',n) = 1$, has a solution. However, as Bill Dubuque's comment indicates, this assumes you're doing integer division to the extent of removing the common factor of $d$. Note that $a^{-1}$ doesn't exist modulo $n$ in this case. However, $(a')^{-1}$ does exist modulo $\frac{n}{d}$, so a possible interpretation would be $\frac{c'}{a'} \equiv c'(a')^{-1} \equiv e \pmod{\frac{n}{d}}$.
As for why the multiplicative inverse $b = a^{-1}$ exists modulo $n$ if $\gcd(a,n) = 1$, Bézout's identity states that in such cases there exist integers $x$ and $y$ such that
$$ax + ny = 1 \tag{1}\label{eq1}$$
As such $ax \equiv 1 \pmod n$ so $x \equiv a^{-1} = b \pmod n$. This value must be unique, modulo $n$, because if there was another value $x'$ such that $xa \equiv x'a \equiv 1 \pmod n$, then $(x - x')a \equiv 0 \pmod n$. Since $\gcd(a,n) = 1$, this means that $x - x' \equiv 0 \pmod n \; \iff \; x \equiv x' \pmod n$.
Bézout's identity also shows that if $a$ and $n$ are not relatively prime, e.g., $\gcd(a,n) = d \gt 1$, then \eqref{eq1} becomes
$$ax + ny = d \tag{2}\label{eq2}$$
with the integers of the form $ax + ny$ are always multiples of $d$, so it can't be congruent to $1$ and, thus, $a$ would not have a multiplicative inverse.
Best Answer
Looks like a typographical error as well as some pretty rubbish notation ($=$ result?? No thanks).
There's two conventional (and closely related) ways of interpreting mod. One is as an equivalence relation (congruence mod m) which is kind of how you have written it: $$ a \equiv b \text{ mod } m \quad \text{if $m|(a-b)$} $$ (here, think of 'mod' as being attached to the relation $\equiv$, rather than either of the numbers).
The other way is as an operator on a number, that is $ a \text{ mod } m = b$ where $b$ is the unique number such that $m|(a-b)$ and $0\leq b \leq m-1$. That's how this text is using it (but in that case it is confusing that they use $\equiv$). Note however that in that case $$ 5 \text{ mod } 100 = 5 \text{ mod } 99 \neq 4 \text{ mod } 99. $$ (They do this correctly in the other examples).