Explain this line segment intersection code

I've been reading this article on pathfinding over polygon meshes using line-of-sight, and in the article are several code snippets. One of them, the one I'm trying to understand, is a simple function that determines whether two line segments intersect.

Here is that function from the article (in C#, though that isn't important to me):

public static bool LineSegmentsCross(Vector2 a, Vector2 b, Vector2 c, Vector2 d)
{
float denominator = ((b.X - a.X) * (d.Y - c.Y)) - ((b.Y - a.Y) * (d.X - c.X));

if (denominator == 0)
{
return false;
}

float numerator1 = ((a.Y - c.Y) * (d.X - c.X)) - ((a.X - c.X) * (d.Y - c.Y));

float numerator2 = ((a.Y - c.Y) * (b.X - a.X)) - ((a.X - c.X) * (b.Y - a.Y));

if (numerator1 == 0 || numerator2 == 0)
{
return false;
}

float r = numerator1 / denominator;
float s = numerator2 / denominator;

return (r > 0 && r < 1) && (s > 0 && s < 1);
}

Without any explanation, I assume that the function accepts two line segments, a-b and c-d. I know the first part checks whether the two lines are parallel, in which case they don't intersect. After that, I'm not sure. I suspect the next bit checks whether the two segments are out of range of each other? The code talks about numerators and denominators, so maybe it's dealing with slope comparisons?

In any case, I'd like to understand the math used here for determining line intersections. I know there's an algorithm that this code is following, but I don't know what it is. How does the formula used in the first part determine that the lines are parallel? I know it works, but why does it work? And what does the function do after that? Why?

I'd much rather understand how the code works, and then write my own code from that knowledge, than just blindly copy someone else's code.