I think I can give you at least a partial answer to this. I can not elaborate on how the approval or disapproval actions can change the game direction but I can tell you that if you are trying to romance another companion in the game you want to get their approval in order to be able to romance them. so when you see a heart icon come up during conversations, you simply choose the dialogue for that option. This way you will gain the approval of your object of affection.
Even if you are not romancing, if you choose to many disapproval options with any companion, they can leave your party. I did this accidentally at the start of the game when I was learning the game, and when I went to speak to, I think it was Cassandra, she was angry with me and would barely speak. So, basically you can lose them as a party member if their disapproval gets too high.
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