This Martin dreadnaught as a problem found on many older guitars. The bridge pin holes have been worn to the point that the ball ends of the strings sit below the surface of the bridge plate. This condition occurs slowly over time when the ball ends of the strings are installed below the bridge plate and are forced against the plate rapidly as the guitar is tuned to pitch. When installing strings on an acoustic with bridge pin the string should be inserted through the hole, the bridge pin installed loosely pushing the ball forward, and the string should be pulled up against the top of the guitar while the pin is pushed gently into place. The pin should not be too tight. It is the ball against the bridge plate that holds the string in place, not the pressure from the pin.
When the plate gets worn it can be replaced entirely or the repaired by removing small divots of wood around each pin hole and gluing in new wood.