It's actually a bit of a strange question when you think about it. It's on the same level as why do you support aviation, modern medicine, robotics etc. Once you get over the popular, media accentuated, hangups the benefits of nuclear power are transparently obvious.
Having been brought up anti-nuclear by default, I first really began to change my opinion after completing a course that looked in detail at energy policy options. It quickly became very clear to me that nuclear power had to be an essential part in the mix of any country seriously looking at tackling climate change. Ignoring nuclear, as seemed to be the norm amongst the so-called environmental commentators, was (and remains) pure idealogical drivel.
I guess you could say that for me that the issue of climate change was the trigger that led to the change, and I am certainly not alone in that regard. However, with increased knowledge came a more holistic appreciation of the technology. I now support nuclear power because it is awesome. From its hostorical roots to its promise for the future, nuclear power is just one of the fundamentally greatest things to ever be developed.
At a 'meaningful' level support for nuclear is essentially support for the concept of science and innovation - and seriously, who could be against that! It is the belief that a rational empirical approach will allow humanity to overcome obstacles and create a better society.
But really that's just fancy window dressing for saying that nuclear power is awesome.