Why do we have multiple neurons in the output layer of a neural network?
If you want multiple things out of your network you need multiple output nodes. In the case of multiclass classification you want multiple outputs, one for each class. These represent the probability distribution over the different classes. For binary you can get away with only one output because the other class has 1-P as the probability. You could say the same for multiclass that you need one less, however this is commonly parameterized with a softmax which needs all values for the denominator.

Other cases might be for example one output per pixel, for example for image segmentation.