It sounds like you know that each $A_{(k)}$ (single integer in parentheses) refers to the entire $k^{\text{th}}$ approximation, like so:
The sequences in subscripts, like $A_{j_1,j_2,\ldots,j_k}$, are called addresses. Each $j$ can be $0$, $1$, $2$, or $3$; a sequence of these with length $k$ can be used to uniquely determine one of the $4^k$ line segments that together form $A_{(k)}$. For example, here are $A_{(1)}$ and $A_{(2)}$ with their segments addressed.
Note that each sequence for $A_{(1)}$ has length $1$. More generally, each sequence for $A_{(k)}$ has length $k$. As we follow the curve from left to right, the sequences appear in a natural order. As we move from $A_{(1)}$ to $A_{(2)}$, the sequences extend in a natural way. In general, any line segment in $A_{(k)}$ with address $(j_1,j_2,\ldots,j_k)$ will decompose into four more line segments in $A_{(k+1)}$ with addresses $(j_1,j_2,\ldots,j_k,0)$, $(j_1,j_2,\ldots,j_k,1)$, $(j_1,j_2,\ldots,j_k,2)$, and $(j_1,j_2,\ldots,j_k,3)$. For example, if we move to the level $A_{(3)}$, the line segment from $A_{(1)}$ with address $\{2\}$ will have decomposed to the following:
Ultimately, an infinite sequence determines a unique point on the curve. This defines a map from the set of all sequences to the curve. We can also place a metric (a notion of distance) on the set of sequences with the property that the map from one to the other doesn't distort distance too much. It turns out that some computations in the sequence space are simpler than in Euclidean space, where the curve lives, but the correspondence between them allows us to transfer results from one to the other.
