I work for Anthem Blue Cross and Blue Shield so I have a bit of experience in this area.
In general, if your employer covers you while overseas and you are injured/sick it goes something like this:
You find a hospital/doctor (Anthem has a 1-800 call in line which can help you locate one if your overseas and don't know where one is)
Then, depending on the policies of the hospital/doctor you are seeing you may need to pay up front. Other physicians/hospitals have no problem filing the charges to your insurance for you.
In the case that you need to file the claim yourself you simply copy all the reciepts and documents and fill out a brief form (not complicated, just basic stuff.. your name, insurance number, where you were seen etc). Mail or fax that to the insurance company and they will reimburse you.
Like the previous posters stated, each policy is different because each employer has specific rules on how they want things to be covered. Don't hesitate to call your customer service and ask all these questions, they should have a wealth of information!