Comprehensive explanation of conservative rasterization algorithm
I'm not sure what exactly you mean by 


  how to visualize a plane that use w value as one of this coordinates


but here's a sketch that will, hopefully, clarify this sentence:


  a line through each edge of the input triangle as a plane in homogeneous (x c , y c , w c )-space


I've used uppercase letters for clip-space, and lowercase letters for window space.



Let's say AB is an edge of your original triangle.
It projects to an edge ab in window coordinates.
Point A is defined using [XA, YA, ZA, WA] coordinates. After projection it will have coordinates [xA, yA] = [XA/WA, YA/WA] and the value ZA/WA will be used to determine which object is closer to the camera at this point.
So your [X, Y, Z, W] always turns into a [X/W, Y/W] in the end. You can see that Z doesn't play any role in where the point is going to end up on the screen. So we just throw it away for now and interpret the remaining [X, Y, W] as 3D coordinates of the points.

Now we have point A = [XA, YA, WA] and B = [XB, YB, WB].

The projected point a will lie on the line that connects [0, 0, 0] and A.
Point b will lie on the line [0, 0, 0] and B. Three points define a plane, and a normal to the plane defined by 3 points can be found as a cross product of two vectors in this plane (OA and OB, for example). 
"For a convex polygon (such as a triangle), a surface normal can be calculated as the vector cross product of two (non-parallel) edges of the polygon."
More on why in Wikipedia here and here.

Generally this moves your problem from 2D domain in the projected coordinates into a 3D domain of not-yet-projected points. The 2D and 3D are "dual" to each other. For example, if a point lies in the plane of OAB, it for sure lies on the line through ab (and vice versa).

The thing you need to remember is that W is your 3'rd coordinate, not Z! I know, it's very counter-intuitive, but it is what it is, Z only carries depth information, it does not affect position of objects on the screen.

I hope I cleared it up for you a bit.