Note that $T_{\sigma,{\mathbb{Z}}}=\{w\in{\mathbb{Z}}^{p+1}\mid\sum_{i}b_{i}w_{i}=0\}$. The linear isomorphism $\phi\colon{\mathbb{R}}^{p+1}\to{\mathbb{R}}^{p+1}$ given by $\phi(w_{j})=(b_{j}w_{j})$ takes $\sigma$ to the standard simplex

$$\sigma^{\prime}=\{w^{\prime}\in{\mathbb{R}}_{+}^{p+1}\mid\sum_{i}w^{\prime}_{j}=1\},$$

and hence

$$[T_{\sigma^{\prime},{\mathbb{Z}}}\colon\phi(T_{\sigma,{\mathbb{Z}}})]\operatorname{Vol}(\sigma)=\operatorname{Vol}(\sigma^{\prime})=\frac{1}{p!}.$$

Write $T_{\sigma^{\prime},{\mathbb{Z}}}$ as the kernel of $\chi\colon{\mathbb{Z}}^{p+1}\to{\mathbb{Z}}$ defined by $\chi(w^{\prime})=\sum_{i}w^{\prime}_{i}$. Then $\phi(T_{\sigma,{\mathbb{Z}}})=\ker\chi\cap\phi({\mathbb{Z}}^{p+1})$, $\chi(\phi({\mathbb{Z}}^{p+1}))=\gcd(b_{i}){\mathbb{Z}}$, and the exact sequence

$$0\to\frac{\ker\chi}{\ker\chi\cap\phi({\mathbb{Z}}^{p+1})}\to\frac{{\mathbb{Z}}^{p+1}}{\phi({\mathbb{Z}}^{p+1})}\to\frac{{\mathbb{Z}}}{\chi(\phi({\mathbb{Z}}^{p+1}))}\to 0$$

gives as desired

$$[T_{\sigma^{\prime},{\mathbb{Z}}}\colon\phi(T_{\sigma,{\mathbb{Z}}})]=\frac{\prod_{i}b_{i}}{\gcd(b_{i})}.$$

Finally, the first assertion is clear. ∎