Quoting
MiddlePart
A third-party beneficiary clause will not prevent you from engaging an attorney to represent you if in the future you have a claim against the other party to the contract.
A third party beneficiary is someone who receives a benefit from a contract but is not a party to the contract. A classic example might be a person who is designated to receive the payment under a life insurance contract. The contract is between the insurance company and a person, but the payments go to someone other than the person who signed the life insurance contract. That person -- the one who receives the death benefit -- is a third party beneficiary of the life insurance contract.
Whether any particular person or business qualifies as a third-party beneficiary in the particular circumstances can be a complicated determination, and a lot has to do with the terms of the contract and the intentions of the contracting parties.
The significance of being a third party beneficiary is that third party beneficiaries can file a claim against and recover damages from a party to the contract based on that party's breach of the contract, even though the third party beneficiary did not sign the contract.
The purpose of a clause of the kind that you quoted is to say that no-one, other than the parties to the contract or their successors or assigns, has a right to bring a claim against a party to the contract in case of a breach of the contract.