I wish they gave a source for the relationship between using these pronouns and telling the truth.

The explanation is given in the paper, citing another study. Thankfully it's not paywalled (hooray for open access!) As the HN article is a summary of the academic paper [1], it probably omitted the detail for brevity.

Here's what the paper says:

  "The honesty of the status updates written by the participants was assessed
   following the approach introduced by Newman, Pennebaker, Berry, and Richards
   (2003) using LIWC. Their analyses showed that liars use fewer first-person
   pronouns (e.g., I, me), fewer third-person pronouns (e.g., she, their), fewer
   exclusive words (e.g., but, exclude), more motion verbs (e.g., arrive, go),
   and more negative words (e.g., worried, fearful; Newman, Pennebaker, Berry, &
   Richards, 2003)."

[1] http://journals.sagepub.com/doi/full/10.1177/194855061668105...