Ten thousand eyes were on him as he rubbed his hands with dirt; Five thousand tongues applauded when he wiped them on his shirt. Then while the writhing pitcher ground the ball into his hip, Defiance gleamed in Casey's eye, a sneer curled Casey's lip.
This stanza describes Casey getting ready to hit and the pitcher getting ready to pitch.
The pressure continues to mount. Every one of those 5,000 fans has their eyes on Casey. (Do the math—that's like 10,000 eyeballs… give or take a few.) Everyone is cheering. (Here we get more synecdoche that tells us that "Five thousand tongues applauded.")
The description of the "writhing pitcher" gives us the impression that he's, perhaps, a little nervous or uncomfortable to be facing mighty Casey.
Casey, on the other hand, seems as confident as ever. He's "defiant." He's "sneering" at the pitcher.