It suggests an overreaction, as if someone has attacked you verbally even though you might have just been stating an opinion or asking a question.
Here are some examples:
* "I just asked if you wanted to go out tonight, and she jumped down my throat! I didn't mean to offend her."
* "He jumped down my throat when I suggested a different approach to the project."
The idiom is often used humorously to describe someone who is easily offended or prone to overreacting.