Meaning of 様になる


I was reading a story on Hukumusume.

In [this story](http://hukumusume.com/douwa/pc/jap/01/01.htm), there is the
following sentence:

[code]

    怒ったネコは、それからずっと、ネズミを見ると追いかける様になりました。
    
[/code]

[Denshi
Jisho](http://classic.jisho.org/words?jap=%E6%A7%98%E3%81%AB%E3%81%AA%E3%82%8B%09&eng=&dict=edict)
says that `様になる` means "to look good". Here clearly it seems to mean "it came
to this...". Am I right?

