I would venture to say that at least 95% (or even 99%) of all airline tickets sold are non-refundable/non-changeable tickets. Most el cheapo fares have a fee in which the value of the ticket minus the fee is allowed to be applied to a future flight (and then again be subject to the fare rules of that ticket).
You need to locate the fare rules regarding the fare you booked. It will specify what changes (if any) the airline is willing to do.
It seems that you are getting an exceptional deal here if they offered to refund the entire value of the tickets considering your flight is about a month away. I don't consider that an emergency but maybe you have extenuating circumstances. And if that is the case the airline is being even more accomodating to your situation.
The non-refundable/non-changeable issue is INDUSTRY NORM and you have to pay very dearly for a refundable ticket (probably at least twice as much---maybe as much as five times as much as the cheapest fare).
This is the reason that travel insurance exists. If you purchased over the phone you were probably even offered it by the airlines. It seems they all sell it now (at a higher price than you can get on your own) for an extra profit center and maybe even for customer convenience. Travel insurance is very inexpensive, imho.
You are very lucky here---you have a way to get your money back. You didn't do your homework when you bought the ticket and you have a "get out of jail free card." Use this opportunity to learn how the airline industry works so that you won't get caught in a similar situation next time. Hopefully you stay healthy though---I wouldn't wish poor health upon anyone. Feel free to PM me if you'd like. I don't mean to sound rude or mean here---I just was trying to be blunt and honest. The above poster is 100% right on too. Most of the time new members don't get this much attention. Take care and get better! And get your money back!!