Let's start with the most elementary example: projective space $\mathbb P^n$. It's not hard to see that that the number of points on it is always $(q^{n+1}-1)/(q-1).$
Note that this is because $\mathbb P^n$ can be always decomposed into simpler pieces: $\mathbb A^n \cup \mathbb A^{n-1}\cup\dots\cup \mathbb A^0$. Interestingly, something similar applies to all $\mathbb F_q$-varieties. Specifically, the Lefschetz fixed points formula from topology applied to arithmetics gives the following statement for a variety $X/\mathbb F_q:$
There exist some algebraic numbers $\alpha_i$ with $|\alpha_i| = q^{n_i/2}$ for some $(n_i)$ such that the number of points $$\\# X(\mathbb F_{q^n}) = \sum_i \pm \alpha^n_i\quad \text{for any natural } n .$$
Numbers $\alpha_i$ have specific properties and "come from geometry": they are eigenvalues of some operators on $H_{et}(X)$, thus there are as many of them as the dimension of this group; and these groups can often be directly compared to the case of $\mathbb C$.