How does a care agency process bills when the elder has long term care insurance?

That depends on the terms of the insurance policy. Some long-term care policies specify that the insurance company will pay home care benefits only if the care is provided by a licensed home care agency. In that case, the insurance company and the care agency arrange for payments to be made directly to the agency. Other long-term care insurance policies pay home care benefits regardless of who provides the care:  home care agency, independent licensed caregiver, unlicensed caregiver, or family member. With these policies, the insurance company pays the benefit amount directly to the insured person, who then makes his or her own arrangements for payment with the agency or independent caregiver.

If you have this latter kind of policy but would like the insurance company to pay a home care agency directly, ask the agency if their billing office will set up a direct payment arrangement with the insurance company. This can let you skip two sets of paperwork. But it can become a pain in the neck if you decide to switch home care agencies, or to receive some of your care from an independent caregiver and want to use some of the insurance money to pay for it; in those situations, you'd need to get the insurance company to reroute its payments from the original agency back to you or to another agency.