To measure it physically, pretend that all the material on one side is suddenly removed. Then the force that would be experienced is the force we are talking about.
T is a symmetric tensor, called the stress tensor.
To show it's a tensor, we need to establish linearity. Further, we can assume that the first two vectors are at right angles. Since all three vectors are coplanar, we can think of the three forces above as stresses on the faces of a prism as shown in the figure. If we take a prism that is much longer that it is thick, we can ignore the forces on the ends. It now follows from Pythagoras' theorem that the areas in this prism are proportional to the three vectors.