How does using HP filter allows us to estimate long-run values for variables such as output and unemployement?
I've found the answer to my question. By using the HP filter, we are able to extract the long-term component of a time-series, namely it's trend component. Using that, we can then take the difference between the series and its trend component at every observation. This leaves us with a cyclical component; if we are interested in estimating gaps between observed values and potential values or natural rates, this is what we are looking for.
Of course, this is the english explanation of what happens; for a mathematical/statistical approach, I recommend reading the original paper.