To explain it like he did in the video:

He knows that the suns' rays are parallel because the sun is so far away.

So, if parallel beams of light are hitting the Earth at seemingly different angles, then it must be due to the curvature of the planet (he showed this by bending the cardboard map to point the towers away from each other creating an angle).

Because he knows the distance between the two towers, and if you imagine them towards extending downwards into the Earth until they meet each other, the angle was 7 degrees, approximately 1/50th of 360 degrees (a circle). The distance between the two towers can fit into 360 degrees 50 times.

The distance between the towers is 800k, so 800*50=40,000k is the circumference of the Earth.