Why does adding a dropout layer improve deep/machine learning performance, given that dropout suppresses some neurons from the model?
The function of dropout is to increase the robustness of the model and also to remove any simple dependencies between the neurons.
Neurons are only removed for a single pass forward and backward through the network - meaning their weights are synthetically set to zero for that pass, and so their errors are as well, meaning that the weights are not updated.
Dropout also works as a form of regularisation, as it is penalising the model for its complexity, somewhat.
I would recommend having a read of the Dropout section in Michael Nielsen's Deep Learning book (freely available), which gives nice intuition and also has very helpful interactive diagrams/explanations. He explains that:

Dropout is a radically different technique for regularization. Unlike L1 and L2 regularization, dropout doesn't rely on modifying the cost function. Instead, in dropout we modify the network itself.

Here is a nice summary article. From that article:

Some Observations:



Dropout forces a neural network to learn more robust features that are useful in conjunction with many different random subsets of the other neurons.



Dropout roughly doubles the number of iterations required to converge. However, training time for each epoch is less.
With H hidden units, each of which can be dropped, we have
2^H possible models. In testing phase, the entire network is considered and each activation is reduced by a factor p.

Example
Imagine I ask you to make me a cup of tea - you might always use your right hand to pour the water, your left eye to measure the level of water and then your right hand again to stir the tea with a spoon. This would mean your left hand and right eye serve little purpose. Using dropout would e.g. tie your right hand behind your back - forcing you to use your left hand. Now after making me 20 cups of tea, with either one eye or one hand taken out of action, you are better trained at using everything available. Maybe you will later be forced to make tea in a tiny kitchen, where it is only possible to use the kettle with your left arm... and after using dropout, you have experience doing that! You have become more robust to unseen data.