Even for curves, the situation becomes clearer if you also allow affine curves and look at integral points, or more generally, $S$-integral points in number field. If the curve is projective, that's the same as looking at rational points. Also, it's better to look at Euler characteristic $\chi(C)$ (so as to get integer values), so

$\chi(\text{projective curve of genus $g$ with $n$ points removed}) = 2-2g-n.$

Then we have

$\chi(C)=2$ for $\mathbb P^1$, which is projective and has lots of points.

$\chi(C)=1$ for $\mathbb A^1=\mathbb P^1\setminus\{pt\}$ is the additive group, so also has lots of points.

$\chi(C)=0$ means that $C$ has the structure of a group, either an elliptic curve (which is projective) or the multiplicative group $\mathbb G_m=\mathbb P^1\setminus\{pt_1,pt_2\}$. In both cases, the set of integral points forms a finitely generated group, so is often infinite, but much sparser than the $\chi(C)>0$ cases.

$\chi(C)<0$ means $C$ is either a projective curve of genus at least 2, or an elliptic curve minus one or more points, or $\mathbb P^1$ minus at least 3 points. In all three cases, there are finitely many integral points. These three major theorems are due, respectively, to Faltings, Siegel, and Siegel, although the last one (which is solutions to the unit equation) follows for units in number fields from Thue's theorem.

So for quasi-projective curves, there fairly coarse Euler characteristic determines the qualitative behavior of the integral points.

In higher dimensions, one has similar conjectures, where the Euler characteristic is replaced, roughly, by the extent to which the canonical bundle is ample, or anti-ample, or 0, or something "in between". This is all made quite precise in Vojta's conjectures. However, in higher dimensions one only gets statements that apply to all points outside of a proper Zariski closed subset.

Summary: the intuition is that "as the geometry of the variety gets more complicated, the integral/rational points get more sparse."