Seems to have labelled "C" and "-C" in a very misleading manner. Take point A. Draw a tangent, that cuts the curve at point C. That's not the point 2A. That's the point C.
Reflect C across the X axis, and that's point 2A.
So you're not missing anything, the article is misleading at best.
In short, to take the sum of two points A and B, draw a line through them both to find a third point on the curve, reflect that point across the X-axis, and the result (which will also be on the curve) is A+B.
If A and B coincide then use the tangent at A.
The point at (vertical) infinity plays the role of the zero.
Reflect C across the X axis, and that's point 2A.
So you're not missing anything, the article is misleading at best.
In short, to take the sum of two points A and B, draw a line through them both to find a third point on the curve, reflect that point across the X-axis, and the result (which will also be on the curve) is A+B.
If A and B coincide then use the tangent at A.
The point at (vertical) infinity plays the role of the zero.