This is an interesting question. Communication is one of the most important aspects of life. For this reason, effective communication is important. So, anything that helps a person communication well is welcome.
When we put it this way, we can say that reading helps a person communicate for at least two reasons. First, if a person is a voracious reader, then this person will be a lover of words. This will allow a person to choose words precisely and appropriately for any given situation.
Second, a great reader often reads about people, their characters and issues. This creates in a person a greater sensitivity when it comes to others. This, too, allows a person to communicate better.
Reading to construct a more effective response:Sometimes, your communication will be in response to something someone else has already written, in which case reading what has already been written is key to communicating strong feedback. For example, if I didn’t read your question carefully and thought you were talking about how to read instead of how reading impacts communication, my response (my communication back to you) wouldn’t be effective at all because it isn’t what you are looking for. Reading carefully what someone else has written can improve feedback (communication back to the person).
Reading to communicate more effectively in general:Overall, I think there is also merit to the idea that reading can make you a better writer (and since writing falls under communication, communicator). Inevitably, you have reactions to what you read. You might be excited, bored, interested, antagonized, etc. Reading what other people have written and evaluating how you respond to different styles/types of writing helps you understand how to more effectively reach audiences when it’s your turn to write. If you read an article where all it did was list out facts and statistics, you might feel overwhelmed and bored. Because of this experience, you will also know to not do the same to your readers, so you probably won’t make the same mistake of only listing facts. You’ll most likely improve on this technique and thus communicate more effectively.