What does it mean when each individual array is sorted in different array but bear no relationship to each other
What they mean is exactly that: the invariant maintained by the data structure is that each of the arrays $A_0,\ldots,A_{k-1}$ is sorted. Nothing is maintained regarding the relative order of elements in different arrays.

The text probably goes on to describe how to perform the search, insert and delete operations. The search operation is simple: you just search the individual arrays. To insert an element, there are two cases: if $n$ is even, we just add a new array $A_0$ with the new element. Otherwise, choose the smallest $i$ such that $n_i = 0$ (possibly $i=k$). We add the new element to $A_0$, and then merge $A_0,\ldots,A_{i-1}$ to $A_i$ (in this order: merge $A_0$ and $A_1$ to $B_1$; merge $B_1$ and $A_2$ to $B_2$; and so on). Element deletion is more complicated; the data structure is only supposed to support search and insert.

The key idea is that when inserting an element, it is we (the designers of the data structure) who choose where to put it.

The text probably goes on to analyze the amortized cost of these operations. I'll leave these details to the text. (Unless it's an exercise, in which case you should work them out!)