Circuits and Closure Under Reductions
If $A \leq_P B$ then since $P \subseteq P/\mathit{poly}$, also $A \leq_{P/\mathit{poly}} B$. This means that there is a function $f \in P/\mathit{poly}$ such that $x \in A \leftrightarrow f(x) \in B$. Since $f \in P/\mathit{poly}$, for every $n$ there is a polynomial size circuit $C$ that computes $f(x)$ for $x$ of length $n$. The circuit needs to have some mechanism for indicating the output length, which might depend on $x$.
Since $C$ has polynomial size, in particular, there are only polynomially many options for the output length, and so if $B \in P/\mathit{poly}$, these circuits can be connected to the outputs of $C$ in order to compute $A$.