I do not follow the proof HERE on page 21, that the induced map is a fibration. Namely, I do not know why they consider the first diagram with $\Lambda^n_k$ and $\Delta$ on the vertical l.h.s and how it follows by the exponential law in 5.1 that this diagram may be identified with the second diagram on page 21,which has unclear for me the top left corner with $\cup_{\Lambda^k_n \times K}$ .And finally why $j$ is an anodyne extension?
ALSO SEE PAGE 20, HERE
Too long for a comment.. I'm definitely not an expert for this but for the first half of the question, well, it seems to me that the second diagram is really just a "reformulation" of the first.
In the first diagram, a horn in $\Delta^n$ is mapped into $Hom(L,X)$. Further, the $\Delta^n$ is mapped to $Hom(K,X)\times_{\ldots} Hom(L,Y)$. In other words, you have an $n$-simplex both in $Hom(K,X)$ as well as in $Hom(L,Y)$ so that its images under $p_*$ resp $i_*$ coincide. The question is, can you map $\Delta^n$ into $Hom(L,X)$ compatibly with the prescribed map on the horn?
Apparently, the second diagram asks exactly the same question: in the first row, you have the same data --- the $n$-simplex in $Hom(K,X)$ is identified with a map $\Delta^n\times K\to X$ via this "exponential law", the horn map $\Lambda\to Hom(L,Y)$ is identified with $\Lambda\times L\to X$, and it is required that the restriction to $\Lambda\times K$ is "compatible". And the question is the same as before: can you extend it from $\Lambda\times L\to X$ into $\Delta^n\times L\to X$ (so that it is prescribed on $\Delta^n\times K$)?