I tried to answer this question as a first year trying-to-figure-mentoring-out person, and realized I don't really have it well articulated from that standpoint--especially the "how" part. So then I tried to answer it as a former student and go from there. I've been blessed with some great mentors, and that question was a lot easier.
What lessons did (do) I get from my mentors, and how?
I think the biggest one, though certainly by no means complete, is simply how to get inspired and grow up. I probably sound pretty young, which I am, but I do feel
a lot older psychologically than when I started FIRST. (I started significantly younger than my age psychologically for another reason.)
As for how they did it, maybe they're just subtle - those sly dogs - but I feel like they didn't really go into it with a specific structure or even conscious goal. And I honestly don't think it would have worked if they had (any less subtly). I was just looking for role models. Show me what a great project manager looks like, and let it bite me when I get it wrong. (I'm a big believer in learning from failure.) Stand up a be a great engineer and let me work next to you.
So now for the other side. Based partially on what I got,
what am I trying to teach my students?
- Get inspired about something. Do what you love and love what you do.
- Learn, accept, and grow from failure. In fact, do it often and early.
- Learn and grow a little everyday.
- Enjoy solving problems and striving for improvement.
How do I convey this?
I'm fuzzier here. Role modeling as best I can, definitely. Other styles, I'm not really sure. I'm ok at explaining material and do so a decent amount, but that alone doesn't seem to meet the definition. I'm not sure at this point. I am, however, learning--students make excellent teachers.

Ask me in a couple years. (In the mean time, we've got plenty of veterans who really don't need my help anyway.)
EDIT: If anyone's interested, I started a thread asking this question from the student side:
Lessons from Mentors.