Weebly is more likely replacing low-end web designers (ie: the web designers who do projects for less than $5,000). But they are also growing market share- Weebly is cheap enough to get people online who otherwise wouldn't be able to afford a website.
If you are interested, I wrote a blog post examining the ways software is (and isn't) eating web design. http://www.sitebuilderreport.com/blog/how-software-is-eating...
Weebly is pretty solid and the only major one that's free to use with your own domain. It just has a small 'powered by weebly' at the bottom. And it generates standard HTML and CSS that you can then move to another webhost later if you want. It also does an htaccess for redirects if you're moving from a previous host using different tech. The only downside is that you don't get full control over the URLs.
"the only major one that's free to use with your own domain". Nearly every website builder lets you do this. Squarespace, Wix, Yola, Virb etc.
Weebly's base package is free and includes the ability to use your own domain. It's the only one.
Weebly seems to be aimed at mainstream people who think "Boy, I should have website for my wedding/club/team/group/gaming clan." Of course, their goal is make that more and more empowering over time though. If you think about it, what you can do with Weebly was potentially difficult for the average web developer 10-15 years ago -- I'd say that's a leap! Before I started programming, I used front-page and dreamweaver to get started. Eventually, I got more sophisticated as I got more interested in building things but it was empowering to create a website without knowing HTML or CSS.