First, the insurance company incurs costs just processing the claim. If things escalate with the other party, they may even incur legal and court costs.

Additionally, there are some costs that are incurred regardless of fault -- like personal injury protection.

Even defining "who was at fault" is a complicated situation. A lot of people presume that it is as simple as: whoever the police issues a citation to at the scene is 100% at fault. But that's not the way things actually work. The way liability is assigned depends on the state, but in a comparative negligence state you could be proportionately liable for as little as one percent of the fault of the accident. Maybe someone else ran a red light, you entered the intersection on green, and hit them. You could end up sharing some of the cost for that, if it is found that you could have avoided the accident but decided not to.

And even after all of this, if the other party runs out of money, doesn't have insurance, or runs away at the scene, your insurance company is stuck paying the costs under their uninsured/underinsurred motorist coverage.