What, exactly, don't you understand about that example?

Did you actually apply MatrixForm to the results? I thought doing that makes evident what ArrayFlatten is doing here.

To spell things out for this example, which starts with what you may think of as a 2-by-3 matrix each of whose entries is a 2-by-2 matrix: The result of evaluating ArrayFlatten upon it produces what you may think of as a new matrix each of whose entries is a single number rather than a matrix; in the original "matrix" there was a total of 4 rows in the matrix elements, 2 rows from the 1st row of the big matrix, and 2 more rows from the 2nd row of the big matrix; the entries of the 1st row of the result are the successive entries from the first rows of the first row of matrices of the big matrix, etc.

Does it help you if you look at the first Basic Example in the documentation as follows?

 m = {{1, 2}, {3, 4}};

 multi = {{m, m, m}, {m, m, m}}
 multi // MatrixForm

 ArrayFlatten[multi]
 ArrayFlatten[multi] // MatrixForm