B737 Captain reported the aircraft rolled forward several inches after the towbar was disconnected and immediately applied brakes to stop the aircraft. The audio transmission from the pushback crew was unintelligible right before the towbar was disconnected.

Narrative

Initial communication audio with pushback crew was good. However; it become increasingly scratchy and difficult to understand as the pushback progressed. The Tug Driver told us we were clear to start engines and we were pulled to Spot XX on the North Ramp ZZZ. The aircraft came to a stop and I heard a garbled transmission from the pushback crew. It was completely unintelligible; so I stated say again?" I heard another transmission that was again unintelligible and I stated "say again or use hand signals." I knew that the push crew wanted me to set the brake but without hearing those exact words I would be only guessing. It was at this point I heard them disconnect the towbar; we had the right engine running and were starting the left engine. I saw tug off the nose start to pull away from the plane and I could see the airplane move forward slightly about 6 inches. I immediately pressed on both brake pedals to arrest aircraft movement. I set the parking brake and I now could see the tug drivers face and it was of surprise. I saw the guide man off the right side of the aircraft change from crossed wands to closed fists indicating set the parking brake. I replied with the open to closed fist gesture of brakes set. I then saw the tug pull further away from the aircraft and saw the Ramper with the nose bypass pin in his hand. The tug driver and Ramper both saluted me as did the guide man off the right wing. I saluted back with taxi light and a hand salute and saw everyone go back to the terminal normally. We completed the engine start and checklists and proceeded to ZZZ1; without further issues."