In part, that ability of good players to keep their composure and remain calm is due to their confidence in their abilities. But I think it's more than that. I think good players understand that a round of golf rarely goes according to plan. You will have some good breaks and some bad breaks; you will hit some good shots and some poor shots. But good players know that, if they just stick with it, rarely will a round end in a disaster, and often a round started badly can end well. And it is the finishing of the round is all that counts.
Bobby Jones pointed out that the best advice he ever received was from Harry Vardon, who said, "No matter what happens, keep hitting the ball." That needs to be our mindset when we set out to play every round. We need to be determined to keep plugging away through the good times and the bad until we hole the final putt. If we really throw ourselves into the game; and manage to play one shot at a time--giving every shot our full attention--we will finish our round tired from the effort, but content in knowing we gave it our best shot. And that, after all, is all any of us can do. In fact, accepting that all we can do is the best we can with what we have to work with on any given day, should help us weather the ups and downs inherent in playing golf.
Some of my best rounds have started with a bogey. You just never know what can happen if you can remain calm and be firmly resolved to keep on trying--to keep on hitting the ball--no matter what happens out there.