The end of the story tells us much of human pride in terms of results. If we let our goals and ambitions get the better of us, we can likely destroy that which matters to us the most.
"Doodle!" I screamed above the pounding storm and threw my body to the earth above his. For a long time, it seemed forever, I lay there crying, sheltering my fallen scarlet ibis from the heresy of rain.
It can lead to the most dramatic moments of terrible failure and pain.
Referring specifically to other places in the text, the brother in particular began to believe himself invincible in terms of what he could do with Doodle:
Once I had succeeded in teaching Doodle to walk, I began to believe in my own infallibility, and I prepared a terrific development program for him, unknown to Mama and Daddy, of course. I would teach him to run, to swim, to climb trees, and to fight.
To see no end to one's own capability is a dangerous thing, especially when going against the advice of an expert. Sure, Doodle had beat the odds and expectations before, but that didn't mean it would happen every time. Humankind, no matter how capable, is all bound by mortality.
One of my favorite quotes about pride in this piece comes directly from the author:
All of us must have something or someone to be proud of, and Doodle had become mine. I did not know then that pride is a wonderful, terrible thing, a seed that bears two vines, life and death.
To think of pride metaphorically like this as a seed with two vines we realizes two forces might be fighting with each other over our abilities and choices within us. To be prideful can be both bad and good. Under certain circumstances, like when doing school work, it is important to take pride in your work. Under others, like when showing someone how much more attractive you are than them, pride is ugly.
Hope these thoughts help.