In Silverman's Arithmetic of Elliptic Curves (page 95), it says
Letting $T \in E$ [an elliptic curve] with $[m]T' = T$, there is similarly a function $g$ satisfying $$\text{div}(g) = [m]*(T) - [m]*(\mathcal{O}) = \sum_{R \in \text{the m-torsion}} (T' + R)-(R)$$
I know that $[m]T$ refers to multiplication by $m$ (addition $m$ times), and (.) refers to the point as a divisor, but what does $[m](T)$ mean?
Silverman's notation here is $[m]^*(T)$ not $[m]*(T)$ and not $[m](T)$.
Here $[m]$ is the "multiplication by $m$" map from $E$ to $E$ and $[m]^*(T)$ is the pullback operator on divisors induced by $[m]$. For a separable isogeny $\phi$, then by definition (earlier in the book) $\phi^*(P)$ is the sum of $(Q)$ over the $(Q)$ for which $\phi(Q)=P$. So $[m]^*(T)$ here is the sum of the $(U)$ with $[m](U)=T$. These are the $T'+R$ for one fixed $T'$ with $[m](T')=T$ and $[m](R)=O$, as stated in the given formula.