日本語 が話せるようになりたいんです = I want to be able to speak Japanese. Since this refers to you "becoming able to speak", you must use the ～ようになる　construction. It is very important in Japanese, so learn it! 英語できるようになりたい。= I want to speak English. 全部わかるようになった。= Now I understand everything. Think about it. It all makes sense.

I want to become a good speaker of Japanese.

This construction ～ようになる also works. 英語を話せるようになるだろう。= You will be able to speak English. すぐに彼はその仕事ができるようになるだろう。= He will be able to do the work soon. その新しい機械が用いられるようになるだろう。= The new machine will be in use.

よう　= good 話せる　= can speak なる = to come. (the last u is changed to an i in this form to use it in this sentance なり...) たい　= wants んです　= the "n" makes it more like you are "explaining"