It's not a disrespectful term by any means. 'Gotcha' is quite a common term in spoken English, and it's much more conversational than saying "I understand". Don't be offended, it's fine :)

Perhaps you're worrying about two different meanings of "gotcha." When it is said in a normal tone of voice it's not at all disrespectful; it means "I understand," and it's friendly. "Don't worry about the road signs. It will say 'Governor Thomas E. Dewey Thruway' but that's just another name for the 'New York State Thruway.'" "Gotcha." If it sounds friendly, it's friendly. However, in an argument or debate, when someone tricks someone into saying something that can be proved wrong, they might say "Gotcha!" in a crowing, arrogant way. In this case is means "Ha ha ha! I tricked you! You fell into my trap!" and of course it is unfriendly. "Clock A is never right, clock B is right twice a day. Which is better?" "Clock B, of course." "Ha ha ha! Gotcha! Clock A is a minute slow, so it's never right, and clock B doesn't run at all--so it's right twice a day!"

You've no reason to feel insulted if someone replies with 'Gotcha'. Conversation in English-speaking countries is often less formal than conversation in many Asian countries, and informal responses like this are common, even among strangers. I'm sure that the American person was simply answering you in a friendly relaxed way, and wasn't disrespecting you at all.