The bit about the unforgiving minute is a reiteration of this:
If you can force your heart and nerve and sinew
To serve your turn long after they are gone,
And so hold on when there is nothing in you
Except the Will which says to them: ‘Hold on!’
There is a point in every endurance contest that could be called "the unforgiving minute". Coincidentally (or more likely not) a minute is about how long it takes to flood your body with lactic acid during high intensity exertion. Lactic acid is a by product of anaerobic exercise, and it burns in your muscles if you keep pushing them at a high intensity.
So that line refers to your ability to meet the moment where you have to physically suffer for success. The point where the hard fought race (or game or battle) is determined - the unforgiving minute - demands something of you.
If you want to succeed you need to meet its demands. In a distance race that 60 seconds worth of distance run happens when you push yourself thru the pain and suffering to match or beat your opponents. When you keep going even tho it hurts. If you can't do it you drop off from the leading group. Doing it doesn't necessarily mean you'll win the distance race but you will match it with the best runners and be able to hold your head high.
It takes courage and determination to resist your bodies cries to stop running and let it deliver excess oxygen to your system. You don't have to stop either, you just have to lower the intensity enough to allow your body to provide oxygen to the muscles that are crying out in pain. Usually that happens to be just enough loss of intensity to lose touch with the race.
The reality is only you will ever really know if it was truly sixty seconds worth of distance run or just sixty seconds worth of pained jogging. So its also a call to judge yourself harshly but fairly. To know yourself whether you gave all you had to give or not and base your own internal assessments on an accurate look at how you perform.
The poem is advice on how to live well. But its not wishy washy. Its specific on the details that can help us live with integrity in a world full of compromise.
The courage and fortitude that filling the unforgiving minute with 60 seconds worth of distance run require will serve you well in the rest of your life.