If you have a category $C$, then you can consider the category of functors $Func(C,Sets)$ from $C$ to sets. This comes with the Yoneda functor, $C^{op}\to Func(C,Sets)$, and is a generally useful thing to think about.
In $(\infty,1)$-category land, you want to start with an $(\infty,1)$-category $S$, and build the $(\infty,1)$-category of functors $Func(S,Spaces)$ from $S$ to spaces.
Lurie is describing one way to do this. Given a quasicategory $S$, he produces a closed model category $(sSet/S, covariant)$ on the slice category $sSet/S$ which "models" this functor category.
The analogy is to the "point category" construction. Given a functor $F:C\to Sets$, you can build a category $P_F$, whose objects are pairs $(c,x)$ where $c$ is an object of $C$, and $x\in F(c)$, and maps $(c,x)\to (c',x')$ are $f:c\to c'$ in $C$ such that $Ffx=x'$. Such a thing comes with a forgetful functor $U:P_F\to C$, and one can produce an equivalence of categories $$ Func(C,Sets) \qquad \Leftrightarrow\qquad (\text{certain full subcategory of $Cat/C$}). $$ The subcategory is that of functors $U:D\to C$ such that given $f:c\to c'$ in $C$ and $d\in D$ such that $U(d)=c$, there is a unique $g: d\to d'$ in $D$ such that $U(g)=f$.
So, fibrant objects in Lurie's covariant model category for $sSet/S$ are supposed to look like left Grothendieck fibrations, and this model category should model "$Func(S,Spaces)$".
Edit: originally, I said that the certain full subcategory was that of "left Grothendieck fibrations", but I don't think that's right. You can play the above story a bit more generally, with $Sets$ replaced by $Groupoids$; the functor $P_F\to C$ corresponding to a (pseudo)functor $F:C\to Groupoids$ is a typical example of what's called a "left Grothendieck fibration" (or "fibered groupoid"), consult nLab for details.