The American Secretary of State asks the authorities of other countries to allow the passport holder to enter without problems and to give him or her help and protection if it is needed.
My British passport has almost identical wording to the American. "Her Britannic Majesty's Secretary of State requests and requires in the name of Her Majesty all those whom it may concern to allow the bearer to pass freely without let or hindrance, and to afford the bearer such assistance and protection as may be necessary."