What's "boy" here?

What does "win [someone] over" mean? Could you give me some examples?

To "win someone over" means to make someone on your side or to prefer/like you. For example, if you won someone over to your team, it means that you were able to convince them to join your side. Ex: I won my boss over through good performance at work. Ex: I won over the girl I liked by being nice to her and helping her with her studies.

Is there any difference between saying "back home" and "at home"?

No, there is not much of a difference between "back home" and "at home". Both express the meaning of "at his home". However, "back home" usually indicates that the subject is away from home while "at home" does not always imply this. In this case, however, both can be used interchangeably since they are speaking about him while showing that he is at home. Ex: How is your family back home? Ex: Are you at home right now?

Is this word "heartwarming" made by combining "heart" and "warm"? What's it mean?

Yeah! That's right. "Heartwarming" combines "heart" and "warm." It means that something is emotionally touching or uplifting. It makes your "heart" feel "warm" with good feelings. Ex: That movie was so heartwarming. I loved the scene at the end. It made me cry. Ex: My friend wrote a very heartwarming letter to me.

What does "aftershave" mean? A cosmetic, perhaps?

You're correct! It's a skincare product similar to a lotion that is applied to the face after shaving. It's usually scented and quite light in texture, and is meant to moisturize the skin after shaving. Ex: My boyfriend's aftershave lotion smells really good. Ex: If I don't put on aftershave after shaving, my skin gets really dry and red.

What does "fuzz" mean? Is it slang?

Here, "fuzz" is slang for "police" or "policeman!" "Fuzz" can also mean fluffy hair. Ex: The neighbors called the fuzz with the music was too loud next door. Ex: Watch out for the fuzz when you drive home. Ex: I love the fuzz on this blanket.