Why Does A Person Get A Good Feeling If And When They Inspire Another Person?

I would say it feels like intuition.  A feeling like you explained when you help someone or give service to someone.  You are giving yourself recognition that you did something good and helped someone.  Knowing that you did a good thing, I would say.  Have a great day.  Try helping someone once a day, doing good for someone, even just opening a door for them.  If you go the extra mile, then maybe they will for someone else, and it will make someone else's day too.  Keep it going along.