Use this phrase when you want to answer "no" to someone's question, but you're not 100% sure that "no" is really the correct answer.

Simply answering "no" makes it seem that you're completely sure that "no" is the correct answer. But if you answer "I don't know", it seems like you don't have any answer at all. "Not that I know of" falls in between. Another example:

A: Have any customers complained about the new application form?

B: Not that I know of.

Another version of this phrase is "none that I know of". For example:

A: Are there any good Indian restaurants around here?

B: None that I know of.