$Set^+_\Delta$ denotes the category of marked simplicial sets. Def. 3.1.0.1
It is claimed that this category is cartesian closed. The tensor in this monoidal category should be $$ (X,E) \times (Y,F) = (X \times Y, E\times F)$$ But what would be the hom-space $Y^X$?
An $n$-simplex of $Y^X$ should simply be an element of $\hom(\Delta^n, Y^X)$ where $\Delta^n$ is given the trivial marking, so by adjunction, an element of $\hom(\Delta^n\times (X,E), (Y,F))$.
In particular an edge is a morphism $\Delta^1\to Y^X$, and it is marked if it's in the image of the restriction $\hom((\Delta^1, I), Y^X)\to \hom(\Delta^1, Y^X)$, where $I$ marks all edges of $\Delta^1$.
That gives you a description of what $Y^X$ must be, now you have to check that it does provide an inner hom.