Why was the phrase taxation without representation so important?

"Taxation without Representation" was one of the base complaints that the American Colonists had with their British rulers.

Most Americans were British citizens, and the Colonies in general saw themselves as extensions of the British homeland, not merely "part" of the Empire. As such, they viewed themselves as having all the rights and responsibilities that normal British subjects had. Thus, the idea that the British Parliament could pass laws and taxation without having any American MPs sitting in Parliament was offensive, in the same way that any British person living in (say) Liverpool would have expected an MP to represent them.

The catch phrase "No Taxation without Representation" neatly summed up this idea that Americans saw themselves as part of a government as a whole, and expected that government would at least allow them a democratic say in it. That the British government would not allow the Colonists such representation helped fuel the idea that it was not a legitimate government for the Colonies; thus, if the British government wasn't a legitimate government, the Colonists must provide their own one. Q.E.D. - a revolution was justified.