In Schlick's 1994 paper, "An Inexpensive Model for Physically-Based Rendering", where they derive the approximation, the formula is:
$$F_{\lambda}(u) = f_{\lambda} + (1 - f_{\lambda})(1 - u)^{5}$$
Where

So, to answer your first question, $\theta$ refers to the angle between the view vector and the half vector. Consider for a minute that the surface is a perfect mirror. So:
$$V \equiv reflect(V')$$
In this case:
$$N \equiv H$$
For microfacet-base BRDFs, the $D(h_{r})$ term refers to the statistical percentage of microfacet normals that are oriented towards $H$. Aka, what percentage of the incoming light will bounce in the outgoing direction.
As for why we use Fresnel in a BRDF, it has to do with the fact that a BRDF by itself is only a portion of the full BSDF. A BRDF attenuates the reflected portion of light and a BTDF attenuates the refracted. We use the Fresnel to calculate the amount of reflected vs. refracted light, so we can properly attenuate it with the BRDF and BTDF.
$$BSDF = BRDF + BTDF\\
$$
$$\begin{align*}
L_{\text{o}}(p, \omega_{\text{o}}) &= L_{e}(p, \omega_{\text{o}}) \ + \ \int_{\Omega} BSDF * L_{\text{i}}(p, \omega_{\text{i}}) \left | \cos \theta_{\text{i}} \right | d\omega_{\text{i}} \\
&= L_{e}(p, \omega_{\text{o}}) \ + \ \int_{\Omega} BRDF * L_{\text{i, reflected}}(p, \omega_{\text{i}}) \left | \cos \theta_{\text{i}} \right | d\omega_{\text{i}} \ + \ \int_{\Omega} BTDF * L_{\text{i, refracted}}(p, \omega_{\text{i}}) * \left | \cos \theta_{\text{i}} \right | d\omega_{\text{i}}
\end{align*}$$
So, in summary, we use $D$ to get the percentage of light that will bounce in the outgoing direction, and $F$, to find out what percentage of the remaining light will reflect/refract. Both these use $H$, because that is the surface orientation that allows a mirror reflection between $V$ and $V'$