The sequence that the question is asking about appears at the end of chapter 6. Ender is playing the game, and he is genuinely sick of the game. He fully believes that the game is rigged to make him lose no matter what choice he makes. Despite knowing this, Ender still drinks from glasses and dies over and over again. Out of sheer frustration Ender arrives at the Giant again and kicks the glasses over. The Giant accuses Ender of cheating and tries to swat Ender. Ender dodges the blow and proceeds to dig into the Giant's eye all the way through his skull.

And instead of pushing his face into one of the liquids, he kicked one over, then the other, and dodged the Giant's huge hands as the Giant shouted, "Cheater, cheater!" He jumped at the Giant's face, clambered up his lip and nose, and began to dig in the Giant's eye. The stuff came away like cottage cheese, and as the Giant screamed, Ender's figure burrowed into the eye, climbed right in, burrowed in and in.

Ender "wins" this part of the game because his actions allow him to arrive in "Fairyland." Ender immediately turns the game off because he believes that his actions prove he is like Peter.

He hadn't meant to kill the Giant. This was supposed to be a game. Not
a choice between his own grisly death and an even worse murder. I'm a murderer, even when I play. Peter would be proud of me.

This sequence possibly shows a few different things about Ender. First, it shows that he is capable of getting frustrated and lashing out in anger. Ender is a fairly cool, calm, and collected character through many different scenarios, but this particular sequence is another example of how Ender is capable of great violence. We saw it in chapter 1 with Stilson. The sequence also shows that Ender isn't willing to accept an impossible situation. He believes the Giant's game can't be beat, yet Ender returns over and over again to play. This shows perseverance on his part, but it also shows that he's unwilling to accept the situation as impossible. His actions against the Giant might be out of frustration, but they do show creative thinking as well.

The “Giant’s Drink” puzzle tests a student’s ability to think outside the obvious choices in making decisions.  After several tries, Ender tries not choosing either drink, but instead attacks the Giant itself, and finds success.  By attacking the giant’s eye, Ender accidentally kills the Giant, not the desired result, but it does worry Ender, and he calls himself a murderer.  The important point here is that Ender, when confronted with an “impossible” choice, creatively finds a third solution.  This ability is an important aspect of his training because beating the Buggers will take creativity.  His act gets him into “Fairyland.”  When the game progresses, Ender finds that thinking creatively will get him a long way, but that his “violence” is another “tool” he must use, but with intelligent decision-making.