At the binomial point process of 2-D, there is the fallowing the distance PDF between a point and n-th nearest points from the point.
$f_{R_n}(r)=2/R\times B(n+1/2,N-n+1)/B(N-n+1,n) \times \beta(r^2/R^2;n+1/2,N-n+1)$
where $β(x; a,b)$ is the beta density function. $B(a,b)$ is the beta function. $N$ is the total number of points and $R$ is the point distribution range.
I add all these functions as follows.
$\sum_{n=1}^{N} f_{R_n}(r)$
So, the $\sum_{n=1}^{N} f_{R_n}(r)$ for $0<r<R$ is the same as the Linear function $f(r)=2 N/R^2\times r$.
I don't know why.
By your link, $R_n$ is the distance to the origin of the $n$-th nearest point from the origin. Notice that \begin{align*} f_{R_n} (r) &= \frac{d}{dr} \ P[\textrm{dist}(0, \text{$n$-th nearest neighbor}) \le r] \\ &= \frac{d}{dr} \ P[\Phi(B(r)) \geq n]. \end{align*} Here, I am using the notation from the paper you linked, in which $\Phi(V)$ is the random variable that counts the number of points that are on $V$. Summing on $n$, we get \begin{align*} \sum_{n=1}^N f_{R_n} (r) &= \frac{d}{dr} \ \left( \sum_{n=1}^N P[\Phi(B(r)) \geq n] \right) \\ &= \frac{d}{dr} \ E[\Phi(B(r))] \\ &= \frac{d}{dr} \ \left( N \cdot \frac{\pi r^2}{\pi R^2} \right) \\ &= \frac{2Nr}{R^2}. \end{align*} In the third equality, we used that each point has independently probability $\textrm{Volume}(B(r)) / \textrm{Volume}(B(R))$ of landing inside $B(r)$.