I am not sure how the author of the video implement this effect. But simply watching the video and reading the comments below, I think one can achieve such effect as follows:
First consider the case with only one light source. You can draw a radial gradient texture in advance. (brightest in the center, dark on the boundary and gray in between. you can draw this using Photoshop.) Render such texture centered on the light source position. Then you need to generate some triangles to render the shadow. Suppose the light source locates at A. Given a line segment BC on any polygons, you need to extend line AB to a point D far enough (at least out of the screen). Also extend AC to point E. Then you have a quad BCED (or two triangles BCD and CDE). Render this quad with black color. Do the same thing to all line segments of all polygons. That's the shadow. And you get the effect for a single light source.
For multiple light sources, you can render the effect of each light source to a Frame Buffer Object (FBO). And the final result is simply averaging the FBOs. (Actually you can use only one FBO to accumulate the color value for each light source. And in the final pass you just need to divide it with the number of light sources).
I think that's a simple way to achieve the 2D light effect.