This can answered without any complicated maths.
It can be related to the following: Imagine you have N$N$ marked cards in a pack of m$m$ cards and shuffle them randomly. What is the probability that they are all at least distance d$d$ apart? Consider dealing the cards out, one by one, from the top of the pack. Every time you deal a marked card from the top of the deck, you then deal d$d$ cards from the bottom (or just deal out the remainder if there's less than d$d$ of them). Once all the cards are dealt out, they are still completely random. The dealt out cards will have distance at least d between all the marked cards if (and only if) none of the marked cards were originally in the bottom (N-1)d$(N-1)d$. The probability that the marked cards are all distance d apart is the same as the probability that none are in the bottom (N-1)d$(N-1)d$.
The points uniformly distributed on a line segment is just the same (considering the limit as m →∞$m$$\rightarrow∞$). The probability that they are all at least a distance d$d$ apart is the same as the probability that none are in the left section of length (N-1)d$(N-1)d$. This has probability (1-(N-1)d/L)N$(1-\frac{(N-1)d}{L})^N$.
Integrating over 0≤d≤L/(N-1)$0$$\le$$d$$\le$$\frac{L}{(N-1)}$ gives the expected minimum distance of L/(N2-1)$\frac{L}{(N^2-1)}$.
