Old-Fashioned and Grandiose
In Tortilla Flat, the characters bust out thees and thous like they're in the King James Bible or a Shakespeare play. It makes post-World War I Monterey seem like a weird, kind of exotic place. And it's not just vocabulary that makes you feel like you're in Back to the Future IV: it's the whole structure of the novel, which tries to make these simple, ordinary characters into holy, beautiful beings.
So what's up with all this weird language? Well, for one thing, Steinbeck was trying to capture a difference that exists in Spanish that we don't have in English anymore: the difference between formal and informal ways of addressing people. In Spanish, you say usted to people in formal situations (like if you're talking to your boss or your king) and tú to people in informal situations (like if you're talking to your siblings or your friends). In English, you used to say you to people in formal situations and thou to people in informal situations; nowadays, everyone just uses you.
Steinbeck uses "thou" to try to capture (in English) that difference in Spanish. So that definitely means that these guys are speaking to each other in Spanish, even though Steinbeck's writing in English. That's something Ernest Hemingway did, too, in For Whom the Bells Tolls: the characters there are supposed to be speaking Spanish, even though Hemingway is writing in English, so they use "thee" and "thou" as well.
Now, as we mentioned in our "Genre" section, Steinbeck has structured his novel after the Arthurian legends. The style of his short chapters, each one relating an adventure and boasting a long title that tells us everything that's about to happen, is modeled after the style of Arthurian legends, which also tend to have brief sections with long titles that tell us everything that's about to happen.
So, to sum up, Steinbeck's style gives the novel an old-fashioned, legendary feel, and it also to bring a grandness and importance to these ordinary folks and events.