My father and I are going to visit the US. Our travel insurance covers medical expenses too but it is from a company in India. If something were to happen to us in the US, how would we invoke the policy? Do Indian firms have offices in countries where they provide a cover?

Indian Insurance Companies have arrangements with companies in US as well as in other countries. You should not worry- you will get service everywhere in the world. Keep your policy handy during travel as it has helpline phone numbers. When you call helpline number you will need the policy number details to proceed further.