You have 3 points, P01, P1 and P2. (Look at the figure right above the equation.) The points create a triangle. This triangle lies on a plane and on this plane are the two vectors T and B. (Tangent and Bitangent) These vectors are more or less random, optimally they are at a right angle, but this may not be the case, the only requirement is that they are not the same.
With all this you can define the edges (E1 and E2) in respect to T and B.
P0 = (U0, V0)
P1 = (U1, V1)
P3 = (U2, V2)
E1 = (U1 - U0) * T + (V1 - V0) * B
E2 = (U0 - U2) * T + (V0 - V2) * B
Going from there:
Now since you know T and B in world space (and the normal N), you can construct the matrix TBN that will convert the normal from tangent space (as in the normal map) into world space (as required from lighting). The following equations basically outline the mathematical principles based on TB matrix. (The normal N is added later to also account for Z value of the normal.)
For arbitrary meshes the tangent and bitangent are vectors along the surface of the mesh. These need to be computed during construction/loading of the mesh. (You can skip the bitangent if you allow to simply do
B = T x N)