Asking why anyone who's building a new website would use Wordpress isn't a very interesting question. A better question is, should I upgrade my current website to Wordpress? The answer is, of course, yes, but why go through the trouble of migrating my entire site over to Wordpress?
The most important quality of Wordpress is that it's popular. It powers 29% of all websites in the world and has a market share among content management systems of 59.8%. A large, dedicated group of users and enthusiasts keep an Open Source project relevant, safe and cutting edge. Unlike other proprietary content management systems that are developed by a handful of programmers and artists, Wordpress is built and maintained by the entire community. The larger the group of people testing and suggesting upgrades and fixes for a product the better that product will become. That is the most relevant reason why you should make the leap from the free site that your cousin built that he doesn't maintain to Wordpress, which is maintained by millions of people all over the world.
Once just a blog, Wordpress has become much more than that over the years. It should be considered a framework at this point. With themes and plugins, anyone can add functionality and styling to an already powerful CMS. There are thousands of free and pay plugins and themes that can accomplish almost anything you need to do for your business or personal site. A side effect of a constantly updating framework is that it changes. That may seem obvious, but its a defining feature of Wordpress itself. The number one rule of working with Wordpress is, don't change the core files. When Wordpress updates it completely swaps all core files for the new files, so any changes made will be overwritten when that happens. That is why Wordpress should be considered a framework. Developers build on top of the core files and not through them. Wordpress has an elaborate action and filter process for hooking into certain aspects of the framework. Instead of changing part of the code, the code sends out alerts to themes and plugins that something can be added or changed here. This allows plugin and theme developers to seamlessly integrate with Wordpress.
Wordpress powers some of the largest websites in the world. If Fortune 500 and large publications use Wordpress we can infer a few things about its efficiency and ease of use. Websites that get millions of views a year need a system that can handle that kind of activity. Of course, there are technical limitations to any piece of software, but one that can be stretched to those kinds of extremes using add-ons and plugins will definitely stand the test of time. Beyond just the outward facing site, the part the public sees, the back-end of Wordpress must be so user friendly that a company of thousands would choose it to maintain a newspaper or a catalog or an event list. If Wordpress can work for companies like that, it can most certainly work for you.
In addition to everything above, we use Wordpress because it takes care of all the boring things for you. It blogs, it has a user management system, it takes care of most of your SEO needs. Instead of re-inventing the wheel dozens of times a year, we get right into the interesting stuff when we take on a project for a client. If someone needs an eCommerce site it's implied that with the checkout process all the other basic things come with it automatically. We wouldn't be able to charge what we do if we had to recreate all of those basic amenities for each project. With that, we owe a majority of our business to the simplicity and power of Wordpress.
If you still aren't convinced, we can give you a million more reasons why Wordpress is powering the Internet today.