Drouet is a member of the Freemasons, which is an actual secret society. We could've picked up on this way back in Chapter One by the narrator's description of his ring, but now it's made explicitly clear.
Drouet goes to a meeting at the lodge and one of his fellow masons (a.k.a. Elks), Mr. Quincel, tells him that the organization wants to raise money to buy new furniture for the lodge, so they're going to put on a play. Quincel wants Drouet's help in finding an actress who could take one of the parts. Drouet agrees to help though he doesn't actually know anyone who could do it.
The whole thing isn't high on Drouet's list of priorities, and a few days later he finds himself in a bind when he receives a letter informing him that the first play rehearsal is Friday night. They need the name and address of that actress he promised them pronto… oops.
Desperate to find someone, Drouet asks Carrie if she'll take the role. She's flattered to death, but not sure—she has no acting experience. But Carrie has secretly wanted to be an actress after having gone to the theater, so it doesn't take much arm-twisting for Drouet to convince her to take the part.
Drouet leaves and Carrie sits in the rocking chair and fantasizes about being on stage.
Drouet drops into the lodge to let Quincel know that Carrie will be taking the part (of Laura). He also decides that Carrie needs a jazzier stage name, so he christens her Carrie Madenda. Quincel gives him the script and he takes it home to Carrie.
She reads through it and finds out that the part is pretty small and requires "suffering and tears." She spends the next day memorizing her lines. When she performs a scene for Drouet, he's super impressed.