$It's a little notationally confusing, so I understand the struggle.
It's that, inside the brackets, you're dealing with the operation as you would for elements of $\Bbb Z$. It might be more intuitive if you use a second notation for the one on $\Bbb Z / \ell \Bbb Z$.
So, for instance, we define addition $\oplus$ in $\zl$ by
$$[m] \oplus [n] \def [m + n]$$
In other words, addition of equivalence classes in $\zl$ gives you the same equivalence class, as you would get if you found the equivalence class of $m+n$ in $\Z$ first.
Similarly, we're now looking at multiplication $\otimes$ in $\zl$ defined by
$$[m] \otimes [n] \def [m \times n]$$
where $\times$ is the usual multiplication in $\Z$. We want to show this is well-defined. You've already verified that $\oplus$ is well-defined, but now you want to endow $\zl$ with a multiplicative operation as well, $\otimes$.
(We just often use the same notation for both since one naturally induces the other, but, as you've seen, it can be confusing.)