Recall that a simplicial set $X$ is called a quasicategory if for every $0<k<n$, every map $\Lambda^n_k\to X$ admits an extension to a map $\Delta^n\to X$. It is good to have a concise definition, but the definition of quasicategories is perhaps too concise and I don't understand the motivation behind it. There are three things I want to clarify:
People say that quasicategories model $(\infty,1)$-categories; $n$-simplices are like $n$-morphisms, so that an $(n+1)$-simplex serves as a morphism between $n$-simplices. But apparently, an $(n+1)$-simplex has $n+2$ faces, so it doesn't make sense for an $(n+1)$-simplex to be a homotopy between $n$-morphisms. How can I interpret an $(n+1)$-simplex as a "morphism between $n$-morphisms"?
If $n$-simplices correspond to $n$-morphisms, then what does it mean for an $n$-simplex to be invertible? Why does the horn-filling condition guarantee that every $n$-morphism, where $n>1$, is invertible?
Why do we omit $k=0$ and $k=n$ from the horn-filling condition? What's the difference between the inner horns and the outer horns? (I understand that the unique extension property for the inner horns characterizes nerves of small categories, but nerves of small categories also have extension properties for the inclusions $\Lambda_0^n\subset \Delta^n$ and $\Lambda^n_n\subset \Delta^n$as soon as $n\geq 4$, so why not require them for quasicategories?)
Can someone give an answer to the above questions? Any insight is appreciated. Thanks!