This post is mostly targeted towards software developers, but still applies to everyone from dog walkers to graphic designers.
You get a deeper understanding of the topics you write about
Writing about a topic ensures that you really understand that topic, no matter if you've worked with it for years or if its totally new to you. As you recite your own understanding, you begin to see it in a new light. Seeing in this new light, a fresh perspective, you may realize that you are missing a few pieces to the bigger picture. This is why people say the best way to learn something is to teach it.
You're advertising yourself
While you sleep, your blog is still up and easily accessible to anyone. You might catch the attention of a potential employer, or find a coding buddy. Any way it goes, having a blog can lead to many professional opportunities.
You get to show off your skills and expertise
In order to give your name credibility, you need to show your work and talk about it. For example, instead of just listing PHP as one of your languages, you can talk about your experiences with PHP and the challenges you've overcame. You can talk about what you did and the reasoning behind each decision.
How to start your own blog
If you're not a web developer or just don't want to design your own blog, there are tons of free blogging sites like Blogger.
If you do want to develop your own blog, you'll need a domain and a host. Personally I recommend Namecheap. How you go about creating your blog is all up to you, depending on what languages you know and what languages you want to learn.