I've never heard this called 'Lawvere's segment' before, but your $L$ is the subobject classifier in the presheaf topos $[A^{\mathrm{op}},\mathrm{Set}]$. In presheaf toposes generally, the subobject classifier $\Omega$ is the presheaf that sends an object $a$ to the set of all sieves on $a$, i.e. the set of subobjects of $\hom(-,a)$. This means in this case that simplicial subsets $S' \subset S$ are in bijection with simplicial maps $S \to L$.
Off the top of my head, I can't be exactly sure what $L$ looks like, but I'd guess it's the constant simplicial set with $L_n = 2$ (the set of truth values) and all maps identities. Then the characteristic map $\chi_{S'} \colon S \to L$ will be the usual $s \mapsto 1$ if $s \in S'$ and $0$ otherwise, while the simplicial subset corresponding to some $\phi \colon S \to L$ will be given by the fibres $\phi_n^{-1}(1)$ in each degree. $S'$'s being a simplicial subset should correspond to $\phi$'s being a simplicial map.
I could easily be wrong about that last paragraph, though. There might be more about this in Mac Lane--Moerdijk.
Edit: I was wrong -- see the comments below.