For a birational morphism $f:X\to Y$ there exists an open dense set $U\subseteq Y$ for which $f$ induces and isomorphism $f^{-1}U\to U$. For a closed subset $Z\subseteq Y$ such that $Z\cap U\neq\emptyset$ the strict transform is defined as $$\overline{f^{-1}(Z\cap U)}\subseteq X,$$ i.e., the closure of the preimage of the part of $Z$ that lies on the part where the morphism is an isomorphism. This is a very important construction and there isn't a universally accepted notation for it.
János Kollár invented the following notation for this: $$f^{-1}_*Z:= \overline{f^{-1}(Z\cap U)}\subseteq X$$ The genius of the notation is that anyone familiar with basic notation in algebraic geometry should understand what it is:
1) As $f$ is birational, $f^{-1}: Y\dashrightarrow X$ exists as a rational map.
2) For any map $g$, it is common to use $g_*$ to denote push-forward of cycles.
The strict transform is really just the push-forward of cycles via the rational map $f^{-1}$.