How do you get a cough?

All coughs are symptomatic.

there alright, they will only soothe your throat though nothing much, but your coughs will come right back if you don't take medicine

coughs tickle your throat causing reaction to where you breathe out the tickle

Simply put, because it sounds like what it is: that's called "onomatopoeic". "Cough" comes from Old English, first recorded in the early 14th century. See the related link below:

you cough to get the mucus out of the tracea (throut)