Source: Growth in U.S. Population Calls for Larger House of Representatives by Margo Anderson.
Bottom line, if we follow the 'cube-root' model, we would now have 588 Representatives based on the 2000 census, instead of the current number of 435 dating from 1910. Sounds about right to me...
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The article you linked says the number 588 is based off of the 1990 census, when it says the population was about 203 million (This census document says the 1990 population was 249 million, so who knows). According to the linked document, the 2000 census counted 281 million Americans, which by the method in question would produce a 655 member house.
To me it seems a matter of common sense that the size of the House should increase to keep up with the rising population. The closer the representatives are to the people, the better representative democracy works. It's simple. Once you pass 1000 things begin to get unwieldy and impractical, but adding 2 or 3 hundred couldn't hurt.
I'm not convinced by the arguments presented here against it.