The problem.
I think how great others are. I look at the success of those around me and want what they have. I am willing to compete but it exhausts me. I am a competitor, and it angers me.
The Zen understanding.
A competition is like a race, no? In life, though, this "race" only ends one way: in your own death. So it is not a race you can ever finish in victory. If you view life as a race, your life will sprint from loss to loss. You will not find happiness.
A Zen Buddhist sees no sense in competition for personal gain. What does it truly matter if you have a little more or a little less? Additionally, competitiveness leads to dislike, envy, and even violence.
That isn't to say you shouldn't try. A Zen Buddhist simply reminds themselves that they try for personal growth and not for personal gain. In a wonderful scenario, a Zen Buddhist tries for the benefit of others' enlightenment.
The Zen application.
In practice, a Zen Buddhist does not race; they walk. They move forward but at a pace bringing them happiness. They go slowly to see the beauty of life. This has the beautiful effect of helping others around them slow down, too. After all, if you see someone happy and content and simple, you are inclined to wonder: could that be me? The truth is: yes! It can be you.
A Zen Buddhist does not dwell on the worldly success of others. A Zen Buddhist takes on a grateful, neutral attitude toward their own accomplishments. To do this, a Zen Buddhist reminds themselves of their own smallness and lack of intrinsic existence.