I am having trouble comparing the usual picture of strict $\infty$-categories as categories with $k$-morphisms for each $k\geq 0$ with their inductive definition.
If one lets $0Cat := Set$, strict $n$-categories are defined inductively by $$ (n+1)Cat = (nCat)Cat. $$ With this definition, objects of $nCat$ have a set of $k$-morphisms for each $k\leq n$, and for each $k>0$ there is a composition of $k$-morphisms.
Moreover, for each $n$ there is a functor $i:nCat\to (n+1)Cat$ adjoining $n+1$ morphisms for $n$-morphisms trivially: $$ \hom(f,g) = \begin{cases} \{\emptyset\}\quad\text{ if } f=g \\ \emptyset\quad\text{ otherwise.} \end{cases}, \text{ for all n-morphisms } f,g. $$
Strict $\infty$-categories are defined by taking the colimit below:
I can't see why does this definition implies the standard description of $\infty$-categories i.e. there are $k$-morphisms for each $k\geq 0$.
To see $\infty$-categories as a colimit of $n$-categories, you need to take some kind of a "completed" colimit. For instance, you can see your functors as functors of locally presentable categories and take the colimit in there. In any case this just amounts to taking the limit in the ordinary sense of the right adjoints, so it's probably indeed more natural to think about a limit.