Having xyz = 0 means that one of the variables is equal to 0. Specifically, x and y can be any real numbers while z is 0, or x and z are any real numbers while y is 0, or y and z are real numbers while x is 0.
This means that the set of points that satisfy xyz = 0 are those that lie in the planes containing the axes: the xy-plane, yz-plane, and xy-plane.
This can be related to the 2-dimensional (cartesian) plane for points satisfying xy = 0 -- either x is any real number and y is 0, or y is any real number and x is 0. The first case simply consists of the x-axis, while the second case, the y-axis. This means that we want all the points the are along any of the axes (x and y).