The left-hand grid represents the domain of T; the right-hand grid represents the range. Try drawing some lines in the left-hand grid; their image will automatically appear in the right-hand grid. Now change the coefficients of T and watch the image change.
The green dot and blue dot represent the basis vectors (1,0) and (0,1) respectively, together with their image under T (the images are of course on the right-hand grid). Note how a and c control the location of the green image dot, while b and d control the location of the blue image dot.
The magnitude of the determinant ad-bc measures the area of the parallelograms in the image grid on the right-hand side; what happens
when the determinant goes to zero? The sign of the determinant measures whether the transformation T is orientation-preserving or
orientation-reversing. (When the determinant is negative - e.g. if one selects the "Reflection" matrix example - try drawing an anticlockwise
loop on the domain. What orientation does this loop map to on the range?)