The spellcheck
attribute is still poorly (if at all) implemented in browsers. For example, when I visit the page you mention in Chrome 18 and type misspelled words into the box “This text area should be checked for mispelled words.” the browser marks “words” as misspelled. Double clicking on some other word makes it indicated as misspelled, too. The explanation is that I had Spanish set as the spellchecking language. Note: This language can be changed by right-clicking on the input are and selecting the spellcheck settings, but the change does not take effect before the page is reloaded.
So check out those settings in your browser. Note that there’s a checkbox there that controls whether spellchecking is in use at all.