This is pretty clear if you work with Lie algebras. Your Borel subalgebra $\mathfrak b$ determines a choice of positive roots: $\mathfrak b = \mathfrak t \oplus \bigoplus_{\alpha > 0} \mathfrak g_\alpha$. The action of $w \in W$ takes $\mathfrak b$ to $\mathfrak b_w = \mathfrak t \oplus \bigoplus_{\alpha > 0} \mathfrak g_{w\alpha}$. Since $w_0$ takes the positive roots to the negative roots, it follows that $b_{w_0}$ is the Borel subalgebra opposite to $\mathfrak b$.