Need of Weighted Mean Squared Error
Weighting MSE is a way to give more importance to some prediction errors than to others in the overall score. This is useful if you are using MSE as a performance metric for your model, especially during the model training (loss function) or validation (hyper-parameter setting).

In the case you cite as example, more importance is given to cases with more clicks. If you use this WMSE as performance metric for validation, your model will tend to be better on cases with high number of clicks, than if you had used MSE.



Just a note on your example: the used squared error ($(predictedClicks - observedClicks)^2$) is an absolute squared error (I would have expected a relative error), and therefore already increases with the number of clicks. So in this case, weighting that way reinforces the performance on cases with high number of clicks.