The idea in backpropogation is instead to mathematically relate a change in output to a change in the parameters. You figure out how much you need to change the parameters to change the output a desired amount. Hence the "back" in the name, since you want to control the output, "steering" it in the direction you want, and to do so you go backwards through the process to figure out how much you need to change the parameters.
Instead of "if I turn the knob 15 degrees the temperature goes up 20 degrees", you want "in order to increase the temperature 20 degrees the knob must be turned 15 degrees".
By comparing the output with a reference, you get how much the output needs to change to match the reference, and by using the backpropagation technique you can then relate that to how much you need to change the parameters.
In neural nets the parameters are the so-called weights of the connections between the layers in the model. However the idea is quite general so here they've applied it to optimizing the size, shape, position and color of (gaussian) blobs, which when rendered on top of each other blend to form an image.
Changing a blobs position say might make it better for one pixel but worse for another. So instead of doing a big change in parameters, you do small iterative steps. This is the so-called training phase. Over time the hope is that the output error decreases steadily.
edit: while backpropagation is quite general as such, as I alluded to earlier, it does require that the operation behaves sufficiently nice, so to speak. That's one reason for using gaussians over say spheres. Gaussians have nice smooth properties. Spheres have an edge, the surface, which introduces a sudden change. Backpropagation works best with smooth changes.
