Just curious about this.
On the one hand, I feel like it's a bit heavy-handed to say "And these interests relate to your research interests because of XYZ." It sounds like you are trying to sell them on something.
On the other hand, it is important that advisors feel your interests align.
Note: These are advisors whose interests do pretty obviously overlap with mine -- it's not a stretch.
What is the consensus on this?
UPDATE: So I had two advisor interviews today (back to back, same dept). The first thing both possible advisors asked me was something like "What are your research interests and why did you apply to X department at Y school to study with my group in particular?" (emphasis mine). So that made it a little easier because they explicitly asked for the link.