Why listening and speaking are inseparable?

Well... they aren't inseparable. There are times when you would only listen and not speak, and there are times, albeit few, where you might speak and not listen... and it would be absolutely appropriate to separate the two during those times.

A President, giving a speech, focuses all of his attention on the speech... at that time, he will not be listening to anything else, but focusing completely on his speech... one way communication, from him to his audience.

A student in a large freshman class listens to a lecture... takes notes, tries to gather all the information possible. May not be able or permitted to ask questions until later, during lab. (Those lecture sessions have hundreds of people and it just isn't feasible.) Gathering that information and just listening is the appropriate thing to do in that situation.

However, most of the time in interpersonal communication, listening is coupled with speaking. Understanding another person and being understood in return is valuable... and one-way communication doesn't work.

Disclosing your innermost self without the other person disclosing anything in return can cause you to retreat... trust needs to be shared, or it is hard to have it at all. Risk is part of love, but a shared relationship requires a shared risk... a shared commitment.