what is init_score in lightGBM?
Omitting the technical details, boosting is a statistical technique where we train many additional weak models to attempt to correct the errors of the previous models we have learned so far. In each iteration of gradient boosting, a new model is trained (usually a decision tree) which tries to fit to the residual of the prediction made by the existing set of models.

The init_score value represents the prediction that you are trying to correct with the first boosting iteration. By default this probably predicts the majority class for all examples or a randomly selected class, but you can input the prediction that was outputted by any other model here if you like.

The first tree that is learned by LightGBM will try to correct the errors of this initial prediction. In the example that you linked, they train a LightGBM model for one iteration, and use the score to continue training another LightGBM model. So in practice, what they have done in this example is kind of pointless because they could have just trained a single LightGBM from the start, but I suppose it is just to illustrate the concept and how it works.