Mastering the use of words and developing vocabulary is a necessary skill to develop because acquiring more advanced skills depends on the comprehension of language. The more you write, the more familiar you become with words and their various definitions, which increases your overall comprehension of language.
People communicate by using words, and writing is a form of communication. Whether it's a story, a news article, a book report or any other written document, something is being communicated to the reader. The better you are with words, the better you will be with language and the better your communication will be in whatever it is you are writing.
Knowing and using proper grammar can also be developed by writing. All written languages require correct grammar to be understood, and when you write it's necessary to use it so that the reader can comprehend what you're trying to communicate. To be good at writing, you must be good at grammar.
Writing can also help with speaking because both are forms of communication and therefore both rely on knowledge of the language in question to effectively communicate in that language. When using proper grammar and a broad vocabulary in your writing, you will most likely begin to use the expanded vocabulary when you speak.