Suppose that B_1 consists of vectors $e_i$, and B_2 consists of vectors $f_j$. Then $\langle Te_i,f_j\rangle = \langle e_i,T^*f_j\rangle = \overline{\langle T^*f_j,e_i\rangle}$. In other words, the (j,i)-element of $[T^*]_{B_2,B_1}$ is the complex conjugate of the (i,j)-element of $[T]_{B_1,B_2}$.