That's a great question! Notice how the circles share a center. Take a look at how Po draws two triangles in the explanation, one from the center of the smaller circle to where the line touches the circle, and the other from the center of the bigger circle (the same center!) to the edge of the circle. Given the triangles have the same two sides, we can conclude that the third side, in this case, is also the same!
That means each side of the chord is the same length, so the radius bisects it.