The exponential type decay is because the deviation between the Tian-Yau metric and the Calabi ansatz is exponentially small, and on the Calabi ansatz model $(-\log|S|)^{\frac{1}{n-1}}$ is precisely harmonic. This is the intimately connected to the freedom to add a constant to $-\log|S|$ in the Calabi ansatz, without affecting the complex Monge-Ampère measure.

To evaluate $\int f_{0}\omega_{TY}^{n-1}$, we take a very large compact set $K=\{\log|S|\leq R\}$, and consider the $R\to\infty$ limit. We have up to exponentially suppressed errors

$$\int_{K}dd^{c}u_{0}\wedge\omega_{TY}^{n-2}=\int_{\partial K}d^{c}u_{0}\wedge\omega_{TY}^{n-2}\approx\int_{\partial K}d^{c}u_{0}\wedge\omega_{Cal^{\prime}}^{n-2}.$$

Computing in the Calabi ansatz model, this is

$$-\frac{1}{n-1}\int_{\partial K}d^{c}\log|S|\wedge(dd^{c}(-\log|S|))^{n-2}$$

which is

$$\frac{1}{n-1}\int_{Y}(dd^{c}(-\log|S|))^{n-2}=\frac{1}{n-1}\int_{Y}(d_{2}c_{1}(L_{0}))^{n-2}=\frac{d_{2}^{n-2}}{n-1}\int_{Y}c_{1}(L_{0})^{n-2}.$$

Taking the $R\to+\infty$ gives the total integral. ∎