Where did the gap come from?

I seem to remember this being on QI...

There is an angle between the two straight edges where the green and yellow triangles join, not one continuous straight edge.

Did Stephen Fry provide an answer?

Someone handed it to me on a piece of paper, so I re-drew the shapes in Excel and then copied, pasted and rearranged them to form the bottom triangle - they are identical.

No they are not.

2:5 is not 3:7 ergo the green and yellow triangles are not similar, so neither of these over-all shapes is actually a triangle.

If you draw a straight line from 0,0 to 13,5 it intersects y=2 at 2/5 * 13 = 5.2 and not 5.0

In the first picture, the un-straight line dips below the true line, and in the second it goes above it. In both cases, these 3 edges make a very slim triangle. The combined areas of these triangles is equal to the area of the "missing" square.

Now my head hurts!

Indeed. It's a sort of optical illusion.
As JJ says, your eyes/brain basically interprets the hypotenuses of the two triangles as the same (i.e. a continuous straight line) even though they are not.
The line isn't straight despite what it looks like and hence there is more space in the bottom arrangement than in the top which is where the white square comes from.

A simple change in gradient between the two triangles...

Illusion? Maybe if you have bad eyesight.

Okay so I had a quick play with the image.



In the top pic, you can see the black line I drew is the true hypotenuse of the triangle. The one created by the green and yellow hypotenuse, the "false" hypotenuse is not a straight line. It's slightly bent inwards. The gap between the false and true hypotenuses (sp?) is what you see in the second pic.

Alternatively, the extra bits of the yellow and green triangle that go over the true hypotenuse could be chopped off and would fit into that square gap!