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to reach the point where ...; to come to ...; to come to be that ...; to start to ...; to begin to ...
「様になる」 has another reading with different meanings.
日本語が話せるようになった。
I have become able to speak Japanese.
漢字が読めるようになります。
You will become able to read kanji.
野菜を食べるようになる。
I will start eating vegetables.