what does it mean by problem of size $0$?
We're talking about algorithms to sort a list.  Therefore, in this context, a problem of size 0 is a list of size 0, i.e., the empty list.

Quicksort splits a list of size $n$ into two sublists of size $m$ and $n-m-1$.  They sum to $n-1$ (not $n$), because quicksort removes the pivot element: the pivot element is not included in either sublist.  Therefore, the text you have quoted is correct.  If one sublist has size $n-1$, then the other will have size 0 (not size 1).