For example, if we have a recommender system based on some data like MovieLens, where we have a matrix of user / movie pairings, how would reducing the dimensionality of the dataset help with privacy?
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If I understand it correctly, PCA can be explained as creating new dimensions which are combinations of the existing dimensions but which are more efficient at describing/summarizing the data without being redundant. See this question for an excellent explanation.
That being said, the new contrived characteristics may depend on the original ones which are privacy related, but the original identifying characteristics are obfuscated by being combined/convoluted with other independent variables to create the new dimensions.
