There are several reasons for this, the most obvious being that it’s good for branding. Nothing looks more unprofessional than a brand that hosts its website at brandname.wordpress.com. This is because as a general rule, the websites that buy and maintain a domain name tend to be those that are more reputable and more established.
But your domain name is so much more than that. Often, the reason why people start a website in the first place is so that there’s something they have full control over. When you use social networking sites or free blogging providers, or even if you make a living selling on Amazon and eBay, you’re fully reliant on those other sites. If they change their terms of service or the service goes down completely, you’re in trouble.
When you’ve registered a domain name, though, you don’t have to rely on anyone else. That domain name belongs to you and you alone, and you have full control over what you do with it. You can control any associated email addresses, determine which server the site accesses and even sell it or transfer ownership if that’s what you want to do.
Meanwhile, the status of your domain name has a direct influence on your website’s position in search engine results pages (SERPs). Older domain names tend to take priority over newer ones, and at the core of every search engine is a citation model. This means that they determine how influential your website is based on the influence of the links that point towards it - so if you don’t own your domain name, any links that you earn will be giving their value to someone else.