It’s exactly how it works. You calculate what your tax bill will be, and instead of paying it in taxes where the government decides what to do with the money (in theory democratically, in practice it’s different obviously, see point #6), it goes into a charity in your name.
Then you use this charity for multiple things:
Free PR, as you don’t need to use your own money, you use the money that otherwise went to taxes. The headline is X donates $N billion to charity, you look so giving, even though it’s money you wouldn’t have been able to keep any other way.
Your foundation donates to prestigious academic institutions. That’s something that you can parlay into a board seat or at least influence. Now you can decide what this institution will do. In Bill Gates’ case, he used his influence to make sure the Oxford vaccine wasn’t open sourced, but instead licensed. This delayed the response in the developing world by a year or so, and made sure that the pharmaceutical industry made even more money than they even made otherwise. Oh, and Bill Gates privately (on the non-charity side), owns a bunch of pharma stock.
speaking of academic institutions: you buy a fancy building for their economics department. Suddenly the whole field of economics is basically limited to professors teaching trickle down economics. Marx’ analysis of economics is considered fringe, and MMT as well.
your foundation throws parties fundraisers where you get to hang out with important people. Catering, venue, entertainment, etc is paid for by your charity. The people donating to your charity are using their own charities to do so, it’s just one big circlejerk with free money that would’ve gone to taxes instead.
your foundation can donate to politicians or political organisations that will advocate for things you want. The things you want are deregulation, less taxes, etc. This in turn benefits you personally again on the non-charity side.