PCA – The death of me

I was handed a research paper by my adviser sometime ago. The paper covered some aspect of social health networks. The paper has taken my down. It contains a certain process for reaching its results, that included Data Collection, Cleaning of data and then applying Principal Component Analysis on it to do dimension reduction. Well 3 months down the road, i still cannot comprehend the data processing and PCA. It seems PCA in theory is very simple (infact i read atleast dozen papers on it). What is cannot understand is how the paper gets from dimension reduction to the new Dataset and retains the variable names. This in theory is not possible :(. I am still at it, If i dont get any progress done on this in the next two weeks then i am going to go nuclear on this PCA issue 🙁