I am seeing this approach first time in an Electrodynamic book. They have used straightforward formula. From the figure, I could able to prove $$x=r\sin \theta \cos \phi,$$ $$y=r\sin \theta \sin \phi$$ and $$z=r\cos\theta$$
I can see that $\hat{r}=\frac{x\hat{i}+y\hat{j}+z\hat{k}}{r}.$ I can deduce the given expression of $\hat{r}$. But, I am not able to deduce the other two expresions. Why did they define $\hat{\theta}=\frac{\hat{z}\times\hat{r}}{\sin \theta}$ and $\hat{\theta}=\hat{\phi}\times \hat{r}?$ How $\theta$ related to $\hat{\theta}$? How $\phi$ related to $\hat{\phi}$?
We define them as the covariant basis vectors in terms of the spherical coordinates $$q^1 = r, \qquad q^2 = \theta \qquad q^3 = \phi$$
The new basis vectors are tangential to a coordinate line. In other words a line where one $q^\alpha$ is hold variable and the rest of the $q^i$ is hold constant.
The new basis vectors $\vec{g_\alpha}$ are derived through $$\left(\vec{g_\alpha}\right)^i=\frac{\partial x^i}{\partial q^\alpha} \frac{1}{\|\partial \vec{x} / \partial q^\alpha \|}$$ where $x^i$ are the Cartesian coordinates in spherical coordinates.
Using this we can derive $\hat{\theta}, \hat{\phi}, \hat{r}$ which have the property that they are perpendicular to one another.
The same property or vectors can be derived by the way your book defined it.
Since the cross product of two vectors is perpendicular two those two vectors, $\hat{\theta}$ should be clear.
Now, can you see the idea behind $\frac{\hat{z} \times \hat{r}}{\text{sin}\theta}$?