You can write the equation of a plane in R^3 as A x1 + B x2 + C x3 + D = 0. So, in this case, we have (A,B,C) equal to (1,1,1), and D=0. (A,B,C) is the normal

You can write the equation of a plane in R^3 as A x1 + B x2 + C x3 + D = 0. So, in this case, we have (A,B,C) equal to (1,1,1), and D=0. (A,B,C) is the normal to the plane, and D is the perpendicular distance between the plane, and the origin. So, x1 + x2 + x3 = 0 defines a plane that passes through the origin (D=0), with normal vector (1,1,1) and of course (-1,-1,-1), and (2,2,2) etc. So it's a "diagonalish" plane that passes through the origin and "faces" (1,1,1).