I do not think there is anyway to get something quite like the video in Godot, at least not yet. Perhaps with particle trails and some form of particle attraction, which as far as I know is not implemented in Godot 3.0, you can create a similar effect.
You also may be able to write a custom shader that manipulates the vertices of a mesh to curve and displace so it looks somewhat like lighting, but I have little experience with shaders so I have no idea how feasible that is.
I would say your best bet is to make a similar visual effect using a something like the method in this tutorial on YouTube. Granted using this method will need some clever workarounds to get it working with varying distances from target to source.