As the previous answer states, the term “broad” is being used here as a slang term for a woman, in this case a young woman. This book was written in 1967 and was set a little earlier than that. The term was used, in those days, to refer to any mature female (but not to prepubescent girls).
The way to figure this out from the book is to use context clues. If you look at the context, “woman” or “girl” is the most likely meaning for this word. The passage you quote is on p. 14 in my copy of the book. There, Steve is asking Dally if he has broken up with Sylvia again. In the next sentence, Dally replies
Yeah, and this time it’s for good. That little broad was two-timin’ me again while I was in jail.
Since the subject of the previous sentence was Sylvia, and since Dally is directly answering the question, the term “little broad” must clearly refer to Sylvia. This makes it likely that a “broad” is a woman.
The word was used for a few decades on a fairly regular basis in the US. It was acceptable enough to use in Broadway plays like South Pacific, as in the following lyric in the song "Honey Bun."
…and she’s broad, where a broad, should be broad…
However, it was always seen as something of a derogatory term not used by cultured people. The use of the term died out with the women’s movement.
Broad is a slang term for "woman." The person is angry that a girl has been playing him.