"I know where I stand" can mean something similar to "I know my place". The meanings mentioned by other answers are also possible interpretations. Disambiguating what meaning is meant becomes easier if the phrase continues for a few more words:
"I know where I stand on gun control" would mean you know what your position on that issue is.
"I know where I stand with Alice" would mean you know what Alice's opinion of you is (and generally suggests that the opinion isn't favorable). This might be related to the other meaning, in that "I know where I stand with Alice" means roughly the same as "I know where Alice stands on me", although the latter isn't a phrasing I've ever heard. This meaning could also be used with respect to a larger group, e.g. "I know where I stand in the company".
I agree Kevin's comment on another answer that "I know my place" suggests subservience and acceptance of your place, while "I know where I stand" is more likely to be defiant (or perhaps bitter), although this depends somewhat on tone of voice.
In your specific example of "I shouldn't have talked back to him", the two sentences are largely interchangeable, but "I know where I stand" fits slightly better if the reason is due to his personal opinion of you, while "I know my place" is better if it is your relative social standings that are the issue.