The benefit for the patient, as mhoran_psprep says, is that they have less paperwork to deal with. They also don't have to come up with the money; often a patient would otherwise have to pay the money upfront and then have get the insurance company to reimburse them. Another big advantage to the provider saying that they accept the insurance is that now the provider has a relationship with the insurance.
How it works if the provider doesn't say they accept the insurance is that if a patient goes to a provider, and the insurance refuses to pay, the patient is on the hook for the money; any agreement between the patient and the insurance company is between just those two parties. The provider is not part of the agreement, and is not bound by any of its terms. If the procedure is covered but the insurance company claims it's not, the patient can't just say to the provider "Go talk to the insurance company". The provider has the right to recover the money from the patient, and it's up to the patient to sue the insurance company.
If the provider says they accept insurance, on the other hand, then depending on the wording, the patient can have their liability reduced or eliminated in the case of a dispute between the provider and insurance.
For the provider, one of the benefits from accepting insurance is that it's a selling point to patients. Another is that the provider now has a reliable source of payment. Instead of having to deal with billing hundreds of patients, they only have to bill a few insurance companies, and the insurance companies have less of a chance of saying things like "Can you wait until my new job starts in a few months?" or "I'm filing for bankruptcy".