One obvious reason is because, nowadays, universities and funders generally welcome the publicity of an internet audience. If your blog has a decent amount of followers that means that your research and, consequently, the university is getting more attention and may even be picked up by mainstream media. Most likely, this will increase your chance of scoring more funding or perhaps even a promotion or new position, though there may also be some concern regarding academics' involvement with social media.
However, an open presence in the World Wide Web also means that your research is accessible beyond the ivory tower of academia. For me, that's probably the main reason which got me into blogging. In the first place, because my research was very much inspired by blogs and other social media outlets that link needlework and other crafts with political activism and feminist genealogies. The images, comments and posts across the web combined with the theoretical concepts and literary works that I encountered during my under- and postgraduate studies send my mind on the fascinating journey that eventually led to the development of my PhD project. (A detailed post on what that actually is, is soon to follow :-) By putting my own research and thoughts back out there, I hope to become part of an on-going conversation and to receive more insight and inspiration along the way.
On that note, I strongly believe that academia needs to descend from the ivory tower and to make its work more accessible to the non-academic world. Not just in terms of collaborating with businesses, medical facilities or government institutions, but in reaching out to the literal everyman and woman. By restricting the distribution and publication of research findings and processes to academic journals, conferences and a specialized book market, the majority of people outside of academia are being left out, even though much of the research directly deals with matters important and relevant to them. A blog can reach part of these people not only because of its accessibility but also because it is free of any constraints of the trade regarding the format of presentation, language, or citation style.
More though, it forces you to leave the comfort zone of academic jargon and to express your thoughts and concepts in a manner that makes them understandable and accessible to a layperson. From my experience, I find that the more I attempt to explain to a non-expert what it is exactly that I'm doing the easier it becomes for me to clearly articulate just that. A skill that will certainly also help with your academic writing and conference presentations.
And lastly, but to a certain extent also very importantly (at least for me), by committing to blogging about your research on a regular basis you commit to producing a written piece of work regularly. Though this piece of writing is most likely vastly different from anything that will actually appear in your final thesis, I believe the positive psychological effects of this practice are manifold. On the one hand, there is the satisfaction of actually having completed a written piece of work. On the other hand, blogging can become a sort of warm-up exercise before moving on to writing a section in the actual thesis: thoughts get structured, processes and methodologies get explained and after all, it does not need to be perfect!