I'm usually curious about people, such as what they're doing, why they do things.
= I'm usually curious about people, such as what they are doing and why they do things.

Especially when someone seems really upset, I want to ask if they have any problems and make them feel better by listening to their story
= Especially when someone seems really upset, I want to ask if they have any problems and make them feel better by listening to their story. They usually tell me what's bothering them, but sometimes people feel uncomfortable with what I ask.

= They usually tell me what's bothering them, but sometimes people feel uncomfortable with my questions.

Then they just say "Sorry, but I can't tell you now, because I don't feel well. I will talk about it next time when I feel more comfortable.".
= Then they say, “Sorry, I can’t tell you now because I don’t feel well. I’ll talk about it next time when I feel more comfortable.”

It's not a big problems, actually, but I realized it's not good to keep asking about someone's problems.
= It’s not a big problem, actually, but I realized that it’s not good to keep asking about someone’s problems.