Insurance companies are required to settle claims when liability is clear and the legal damages are reasonably ascertainable. The duty to settle is implied by the insurance policy itself, which says the company will defend and indemnity its customer. This contractual obligation has been interpreted by case law and is further governed by regulations and statutes.
Among other things, the insurance company is required to open a claim, investigate the claim, defend the customer, and if possible settle the claim for an amount up to the limits of the policy. The breach of any of these duties might give rise to a claim against the insurance company itself for "bad faith" breach of contract, among other possible causes of action. That area of law is referred to as "bad faith" law.
For a good summary of the related laws of all the states, see Scott G. Ball, The Right and Duty to Settle Third-Party Liability Claims: a 50-State Survey.