Possible meanigs are 1) James describes how someone might argue against his teaching and how he would respond to that person or 2) James describes how someone might support his teaching. If the speaker is supporting James, then the quotation would end at the end of the verse.
The meanings in the abstract nouns "faith" and "works" can be expressed with "believe in God" and "do good things." Alternate translation: "'You believe in God and I do good things.' Prove to me that you can believe in God without doing anything, and I will prove to you that I believe in God by what I do" (See: [[rc://en/ta/man/jit/figs-abstractnouns]])