If you plan to spend a week or more in Taipei, and so have enough time to soak up the atmosphere of the city as opposed to rushing from one attraction to another, this article may be of interest. Some of the reasons listed apply to every part of Taiwan (health care is the same price wherever in the country you go), and some are actually truer outside Taipei than in the capital (the cost of living is much lower in the south than in the north, for instance). But overall it's an excellent summary of why a lot of Western and Asian tourists and expatriates adore Taipei. The accompanying images are superb, too.
The photo above shows the exterior of Taipei's Museum of Contemporary Art.