How to show convolution is associative?

If I interpret your sums correctly, your equality is just different notation. The outer sum is $ab=x$, so given $x$ sum over all pairs $(a,b)$ such that $ab=x$. The inner sum is all pairs $(s,t)$ such that $st=a$. This is exactly the same as given $x$, sum over all triples $(s,t,b)$ such that $stb=x$. Similarly you can rewrite the right side of the equation as the single sum over all triples $(a,s,t)$ such that $ast=x$. Now rename the variables $(s,t,b) \mapsto (a,s,t)$ and you see that both sums are equivalent.