If you have an accident, and it's not your fault, the insurer you are with will usually tell you that you need to claim off the other person's insurer. In your policy booklet, there is a section called Uninsured Loss Recovery. That will give you the name, address and phone number of an external company of personal injury solicitors who will recover the money from the third party insurer for you. Once you pass the claim on to that external company, your insurer no longer deals with it.
The external company will organise the repairs to your car, arrange a hire car while yours is off the road, deal with any personal injury claims, recover any expenses you get as a result of the accident etc, and they do this by claiming from the third party insurer.
If their claim is successful, they claim their fee off the third party insurer, and you pay nothing. However, the reason you pay your insurer for legal protection is so that if their claim is not successful, and the solicitor cannot claim back their fee from the other person's insurer, they don't need to send you a bill for their services. The insurer would pay their fee because you have got the legal protection.
It's a personal choice, but I have had an accident in the past and the court case went on for 6 months. It was peace of mind to know that I wasn't paying as the legal fees for that amount of work would have been ridiculous.
I know there are adverts on TV for personal accident solicitors, and most of them say that there is no charge, but they will usually ask you to sign a contract so that if their claim fails, you get the bill, so unless you are 100% certain of a successful claim, I would personally avoid them.