Perhaps it will be more illuminating to write
$$ | \hat{n} \rangle = | \phi , \theta \rangle $$
Note that any direction is completely specified by any two angles $\theta, \phi$. You can think of this precisely as $| x \rangle $ except for angular coordinate. Sakurai is just trying to be general/lazy. Also note that this $\hat{n}$ has nothing to do with the radial eigenvalue $n$.
Actually more precisely, you may write, since the coordinates are independent,
$$ | \hat{n} \rangle = | \phi \rangle | \theta \rangle $$
Therefore, since we consider
$$\langle x | \Psi \rangle $$
to be the probability that the particle is within $x$ and $x+ dx$, we may identically regard
$$\langle \theta , \phi| \Psi \rangle $$
as the probability that the particle is between angles $\theta$ and $d\theta$ and $\phi$ and $d\phi$. That is, it is the angular part of the wave function, which can always be expanded in spherical harmonics due to their completeness on the surface of a sphere.
Now since in our case $|\Psi \rangle = | l, m \rangle$ we have that $$\langle \theta , \phi| l,m \rangle = \Psi_{l m}(\theta, \phi) \propto Y_{lm}(\theta, \phi) $$
All we are doing is projecting the angular momentum eigenstates $|l,m \rangle$ into position space. It turns out that the eigenfunctions corresponding to these angular momentum eigenstates are just the spherical harmonics. This is found by writing $L^2$ and $L_z$ in position space and finding the common eigenfunction (which it turns out are the $Y_{lm}$s.