Every auto, homeowner, and umbrella insurance policy has a stated maximum the carrier will pay on a covered claim. That maximum is the policy limit. In Missouri, the auto-insurance minimum is $25,000 per person / $50,000 per accident; in Illinois it's the same. Most policies are written above the minimum.
When a case 'settles at policy limits,' it means the insurer paid the maximum its policy allows. Any additional damages above that amount have to come from the at-fault person's own assets (rare and difficult), an umbrella policy, or your own underinsured-motorist (UIM) coverage.
Demanding policy limits is often the first step in a serious-injury case. The carrier doesn't have to disclose the limit, but a substantive demand with strong damages typically forces the disclosure.