And a practical example of this is checking transformed points against a view frustum. Instead of transforming points into the view space a transposed matrix allows you to transform the frustum into the object space and check untransformed points against it. This works only on non-perspective transforms, of course, but the view transform should not be perspective anyways.
To visualize this, take a simplest case of 2D space and non-homogenous coordinates. A simple frustum would be an angle made by two rays from the origin. You can see that rotating this space is as same as rotating the frustum in the opposite direction (though in this case transposed matrix is the same as inverse) but stretching the space opens/closes the frustum depending on in which direction it pulls its normals.
To visualize this, take a simplest case of 2D space and non-homogenous coordinates. A simple frustum would be an angle made by two rays from the origin. You can see that rotating this space is as same as rotating the frustum in the opposite direction (though in this case transposed matrix is the same as inverse) but stretching the space opens/closes the frustum depending on in which direction it pulls its normals.