In the English language, the phrase "keep me on my toes" is called a "clichè". A clichè is any commonly used slang English phrase with an inferred or implied meaning. Sometimes they are traceable to some popular historical event or historical cultural era, and sometimes they've been around for so long that no one is sure where they come from.
"Keep me on my toes" is a clichè that means "keep me alert, aware, and prepared" so you will not get caught off-guard (off-guard is yet another clichè.
Although I could not find this particular clichè in my dictionary of clichès, I recommend that anyone who speaks English as a second language obtain this dictionary of clichès. I included a citation of the dictionary below, although I did not use it in this reply.