For insurance the distributed ledger solves two big problems.
The first is the transfer of policy and claims data between interested parties. Currently these parties need to keep their own set of records that keep track of what risks they are liable for. Even organisations that use a trusted intermediary keep their own copy of these records. With a distributed ledger there is one golden source of data that everyone uses, and one way that data can be modified (defined in smart contracts), so there is no need to maintain separate systems.
The other big benefit of distributed ledger is frictionless, real time payments that are much cheaper than on traditional payment systems. These combined benefits result in significantly lower operating costs for insurers and the opportunity to introduce new products.