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In C++, a 2D array is essentially an array of arrays, often used to represent a matrix or a grid. 2D arrays are declared with two dimensions, the first specifying the number of rows and the second the number of columns. Easy to visualize as a table, a 2D array is structured in rows and columns, just like a chessboard or an excel sheet.

For example, if we want to declare a 2D array called 'table' that consists of 3 rows and 3 columns, we would write it as:

int table[3][3]; 

Here, we've specifically used the number 3, but if a constant variable like 'arraysize' is defined, you can use it to set both dimensions to ensure consistent array sizes.
In the provided exercise, since 'arraysize' equals 3, our declaration would look like this:

int table[arraysize][arraysize]; 

This declaration sets up our grid-like structure where 'arraysize' indicates both the number of rows and columns.

Whenever we work with 2D arrays, nested for loops become our essential tool. These are simply loops within loops that allow us to iterate over each row and then each column within that row. This is similar to traversing each cell in our table grid one by one.

Let's look at how we would initialize our array 'table' using nested loops:

for(int i = 0; i < arraysize; i++) {   for(int j = 0; j < arraysize; j++) {     table[i][j] = i + j;   }}

This code will set each element of 'table' to the sum of its indices. The outer for loop iterates through each row, while the inner loop iterates through each column of the current row. It's like going through each box of a checkers board, row by row and within those rows, square by square.

Array indexing is how we access elements within our array using their location. In a 2D array, the location of an element is specified by two indices: one for the row and one for the column. The indices usually start from 0 in C++, so the first element is at index [0][0], the next element in the same row but next column is at [0][1], and so forth.

Considering 'table' as our example 2D array, the expression 'table[i][j]' refers to the element in the 'i-th' row and 'j-th' column. In the initialization loop, 'i' and 'j' are the control variables used to index into 'table'. In the provided exercise, each element of 'table' is set to the sum of its row index and column index, demonstrating a direct application of array indexing in array manipulation.
When printing the array, we again make use of indexing within a nested loop structure to access each element and print it, resulting in a formatted output on screen.