I have a question about what explaining unique variance means in regression models and outputs.

I often read in research papers that "We found that outcomes E and F could not simply be explained by individual differences C and D. Each predictor, predictors A and B was uniquely related to one measure of outcomes E and F above and beyond individual differences C and D. ... Taken together, outcomes E and F may be shaped by individual differences C and D, but cannot be explained fully by those constructs."

Does this mean a multiple regression model was computed with A, B, C and D as predictors and E and F as outcomes? And if A and B explain unique variance, does that mean the correlation coefficients for those two predictors are significant? Essentially, what does it mean that predictors A and B was uniquely related to one measure of outcomes E and F above and beyond individual differences C and D?

Here's the DOI to the paper: http://dx.doi.org/10.1037/emo0000927 and the public link. You can look at the first paragraph on page 10 under the "Discussion" section of Study 1 and the "Perceived benefits of IER Interactions" for analysis strategy under Study 1.

Sorry if the question is too abstract. I don't have access to the data nor the regression models. Happy to elaborate more, and I appreciate any input on this!