He's pretty much already given the solution, but I'll give a brief explanation.
When the two collide (not overlap, as that gives 2 intersection points), the point of intersection is going to be the radius length from center of one of the circles. You just have to determine at which angle, which is simple enough to do. Subtracting the two positions will give you a direction vector.
So X2-X1 will give the direction of circle 1 towards the second. If you prefer an angle of direction instead of a vector, ATAN can be used. Plugging that angle into sin/cos will give a normalized direction vector, at which point you can multiple by the radius of the circle plus the originating center coordinate.
In his first example, he's basically using linear interpolation but because the direction (the first part in parenthesis) isn't normalized, multiplying by the radius gives a much larger value. That is why it's then divided by the overall length of the two radii combined.
If curious, I have several algorithms for testing intersections with shapes.