Pick a model $\mathcal{X}$ such that each $\theta_{i}$ is determined by a nef class $\theta_{i,\mathcal{X}}\in N^{1}(\mathcal{X}/S)$. The restriction of $\theta_{i,\mathcal{X}}$ to each component $E_{\omega}$ of $\mathcal{X}_{0}$ is then also nef, and it follows that the intersection number $(\theta_{1,\mathcal{X}}|_{E}\cdot...\cdot\theta_{n,\mathcal{X}}|_{E})$ is non-negative, hence the first assertion. Since the constant function $1$ corresponds to the vertical divisor $\mathcal{X}_{0}$ we have by definition

$$\int_{X}\theta_{1}\wedge...\wedge\theta_{n}=\mathcal{X}_{0}\cdot\theta_{1}\cdot\ldots\cdot\theta_{n}.$$

By [Ful98, Example 20.3.3] this is the same as the intersection number against the generic fiber of $\mathcal{X}$, and this is equal to $\{\theta_{1}\}\cdot\ldots\cdot\{\theta_{n}\}$ by definition. ∎