I don't think this is actually true. It's a nice rhetorical soundbite, but if you count properly (e.g. all taxes all assets owned by Buffett actually pay) there's no way he pays less unless his secretary is extremely exceptional for some reason. It is true that her tax structure is probably different from Buffett's, so if you creatively form your query so that taxes that Buffett predominantly pays are excluded and taxes that the secretary predominantly pays are included, you could probably find a way to justify it, but that would not be a valuable insight.
And of course the whole "pays less", as formulated, is a bit deceptive - his rate of one of the taxes is lower as a percentage, but he actually pays vastly more in actual money (Berkshire Hathaway alone, in which he holds a substantial share, pays billions in taxes every year).
When I got a bit more money, mortgage, stock options, investments, etc. then I started to have options - do I pay this way or that way? Do I invest in this account or that account? Does HDHP/HSA make sense for me or not? Should I sell the certain investment to harvest the tax loss or keep it? Etc. etc. then my asset structure is more varied, so are the opportunities to pay more or less tax. I imagine for really wealthy people (definitely not me, still) there are even more options how to pay less taxes, and they are doing right serious considering them (or paying people to seriously consider them) - if I ever get rich, that's what I'd do.