There was ease in Casey's manner as he stepped into his place; There was pride in Casey's bearing and a smile on Casey's face. And when, responding to the cheers, he lightly doffed his hat, No stranger in the crowd could doubt 'twas Casey at the bat.
This stanza gives us our first real description of Casey, the man himself.
We don't get much physical description, but the speaker makes it clear that Casey is a very confident guy. This is a high-pressure situation and, instead of breaking out in a nervous sweat, Casey is smiling and tipping ("doff[ing]") his cap to his adoring fans. He's relishing the attention.
We also get the sense that Casey has an unmistakable physical presence. Even from the bleachers someone who'd never seen him play (a "stranger") would still know for sure that it was mighty Casey stepping up to the plate.
Take a look at the end of line 21, there. Thayer could have said, "as he stepped up to the plate." Instead, he wrote "stepped into his place [our emphasis]." What's the difference? Well, the phrase "his place" gives us the sense that Casey really owns this spot. It's his.
Literally, we are talking about home plate, but figuratively we are talking about the limelight, the pressure situation with the game on the line. This is Casey's spot. This is where Casey belongs.
The bottom line: Casey has some serious swagger and he loves the spotlight.