Boolean circuit with two inputs and advice input is hard-wired
Just imagine the circuit $D_n(x,\alpha)$ has $n+a(n)$ bits of input, where the first $n$ bits are treated as the first parameter for the machine $M$, and the last $a(n)$ bits are treated as the advice. To forget about this separation and have a simpler notation, we just say $D_n$ accepts two inputs. Now fix the "second parameter" with the correct advice $\alpha_n$, and this induces a polynomial circuit for $L$ with $n$ inputs.