It means "to make an unexpectedly aggressive response to something".
Let's say we're at work, and I'm looking for a file. I come into your office and say, "Petr, have you seen that red file where we keep the list of prepositions?"
If you reply "What do you mean, have I seen the red file with the list of prepositions? Are you accusing me of taking your stupid file? What makes you think I would have seen it anyway, huh?", you have "jumped down my throat".
Or perhaps it's your red file. "Petr, can I borrow your prepositions file, please?" "Borrow my prepositions file? Why do you want my prepositions file? You're always borrowing my files! Why don't you get your own prepositions file?" Again, you have "jumped down my throat".