I have had a personal website in one form or another for 15 years now, while I was still in school. Dating back to when I first starting building websites, in fact, the first website I ever built was my own.
Over this time and my site has taken many forms from a business site selling services to a CV (resume for the American out there) site and a blog - twice. Of all the sites I have managed for myself the types I derived the most benefit from were by two blogs or I should say are my blogs as the latest incarnation of my site is this very blog you are reading now.
1: You develop better ideas
Through having a personal blog and writing at least once a week you become a better writer and a better communicator. As your write about your idea, you gain clarity of the idea itself and allow the idea to develop and progress. This is even true before you press publish and put your idea out for the world to read.
2: Boosts your professional life
I have had times when a potential client has Googled me, not my business name and found my personal site giving them an insight into my interests and my writing style. This lead to a conversation about some shared interests and they are still my client to this day as we have built a solid relationship over time.
Having your own websites gives you a platform you control, to put your best foot forward and show your real interests and abilities.
3: Lots of connections
I have had people reach out to me before about a topic I have written about and we have since collaborated on a project and stay in touch to this day. These connections you develop via your personal website come out of the blue and won't be happening every day especially in the first week of publishing. This is something that happens over time as you start to be seen by a wider audience and have your content show up on Google searches.
4: You expose your self to interesting opportunities
Leading on from benefit 3 you are putting yourself out into the world for people to find you and when you do that you will find you have offers coming in for several collaborations, guest posting on other sites, speaking, this list of possibilities is endless.
5: You have a lot more impact
This is something that comes in time once you have developed a readership.
A presence on the Internet is the modern equivalent of being really good at networking. Having a personal website that you write on regularly allows you to speak to a wider audience and gain momentum for your ideas. An idea written on your website can be seen by hundreds of thousands of people from all around the world, that is far more impact than I will ever have sitting in my home town interacting with a few locals.
6: You can monetise it if you want
This should not be the reason you start a personal website, however, after several years you will have built an audience and have a large amount of content on your site that people keep coming to read. You can monetise this audience in many ways if you so wish.
Overall the value of a personal website that you write on regularly and regularly writing is the key point for gaining any of these benefits. Even while you are sleeping your work is out on the internet and people are reading your thoughts, ideas and words. Giving you an army of little robots working 24/7 to share your ideas with the world giving you all of the above benefits around the clock.