As someone who knows nothing about machine learning and nothing about PCA (well, until now :)), can someone please explain how the two relate to each other? Is one of them a subset to the other, or what?
It happens that having too much different features is not necessarily a good thing, in a phenomenom called curse of dimensionality.
Due to this, we are interested in trying to reduce the number of attributes our algorithm will process. There are two big categories of methods to do that: feature selection and feature extraction.
In feature selection, you try to select the attributes that are the "best" to predict your value. For example, computing the statistical correlation between the attributes and the value you want to predict, and choose those with the highest correlations.
In feature extraction, you create new attributes that are a linear combinations of the original attributes. PCA is a feature extraction algorithm.
In your effort to predict whether a person will follow dietary guidelines for healthy eating you could just assign each activity as its own input to the model. Or, you could apply PCA (and something like varimax factor rotation) and what you might find is that these activities seem to reflect three somewhat separable latent variables that is: physical fitness, competitive athletics and friendship/team based social activity dimension. You now potentially have reduced 50 individual activity measures into 3 dimensions.
Next you would think more deeply about the specific items and combine them in into 3 scales and use the scales as a reduced dimensional input into the predictive model.
Essentially it's distilling your data down to what is most relevant, and thus helps say a classification algorithm work better by only training on the reduced "more manageable" data.