Composition in explicit substitutions
Substitutions form a monoid and they act on terms. We have a choice of writing them as either left or right actions. Sometime in the previous millenium someone decided they act on the right (page 5). A right action should satisfy $a [s \circ t] = (a[s])[t]$, so it makes sense to define the operation $\circ$ that conforms to action on the right. That's all.

You could do entirely the same thing with functions and write them as $x \mathbin{//} f$ (as Mathematica does) in which case you would be seriously tempted to define $x \mathbin{//} (f \circ g) = (x \mathbin{//} f) \mathbin{//} g$. In essence you'd move from the category of sets to its opposite.