I came across this rule in my old notes, but I need an explanation to how it could possibly originate:
The theorem says that for any lattice $L$ in $\mathbb{R}^n$, the order of the quotient group, $\lvert L/aL\rvert$, where $a$ belongs to $\mathbb{R}^n$, is given by $\text{vol}(aL)/\text{vol}(L)=|a|^n$
Any explanation for this rule ?
Try considering the case of $L = \mathbb{Z}^n$ and see if it doesn't clear something up:
If $L = \mathbb{Z}^n$, then $a L$ consists of the points of $\mathbb{Z}^n$ where every coordinate is divisible by $a$, so the quotient $L / a L$ has a total of $a^n$ representatives, namely
$$(0, \ldots, 0), (0, \ldots, 1), \ldots, (a-1, \ldots, a-1)$$
In general, when you multiply $L$ by $a$, you're "thinning out the lattice" by a factor of $a$ in $n$ different dimensions, so $a L$ will be $1/a^n$ the size of $L$, and consequently $|L / a L| = a^n$.
UPDATE: As HSN points out in the comments, apparently the source of confusion is that you have $a \in \mathbb{R}^n$ rather than $a \in \mathbb{Z}$ in the notes. I'm not an expert and I won't swear to it, but surely that's just a mistake -- see my comment below.