When y and z are both 0, that equation reduces to . When x and z are both 0, that equation reduces to . When x and y are both 0. that equation reduces to . So the plane contains the points , and .
The "pyramid" has that as base and the xy, xz, and yz planes as sides with (0, 0, 0) as the vertex.
As far as the form of the answer in the book is concerned, they are probably using the fact that the volume of a parallelpiped, formed by three vectors, u, v, w, at one vertex is the "triple product", . You probably know that the cross procuct of two vectors can be written as a "symbolic" determinant:
The dot product of u with that is
It is not difficult to show, geometrically, that the volume of a "pyramid" (tetrahedron) formed by one corner of a paralellepiped is 1/6 of the volume of the parallelpiped.