When someone who I am following replies to someone's tweet who I'm not following, the tweet that the person who I'm following doesn't show up in my news feed, I can only see it if I go look at that persons name. Why is that and how can I change it?
That's how twitter was designed in mind - since you're not following the person to whom it was replied to, the tweet will not be shown. The only way to change this would be to follow the other person.
Fun fact: During the initials of twitter, this restriction was not around and you were able to see all replies made by a person to another, even if you were not following.