In mathematics, we often write relations between $a$ and $b$ in the form $aRb$. I mean this both in the sense that we write that string to represent an abstract relation, as well as using that form to write expressions with particular relations. In almost every case, these are read as "$a$ [relation] $b$." For a few examples, we have
- $a:=b$, "is defined to be"
- $a\geq b$, "is greater than or equal to"
- $a\in b$, "in / is an element of"
- $a\subseteq$ "is a subset of"
- $a\to b$, "maps to / is mapped to"
- $a=O(b)$, "is big-O of"
Notably, every relation on this list is antisymmetric, so the ordering of $a$ first and then $b$ is important. This list is extremely incomplete, and there are dozens more.
The correct reading of the symbol $|$ is "divides / is a divisor of." When interpreted in this way, $a|b$ aka "$a$ divides $b$" fits this very well established pattern perfectly. Although it might be counter-intuitive to someone who has more experience with arithmetic than mathematics, it's actually a manifestation of a highly standardized pattern.