In the US, our taxes support these museums, as I'm sure they do in Poland. However, many of the museums across United States are for-profit entities. It's a big difference in many regards between much of the rest the world and the United States. Capitalism rules the day here. Much of these entities receive funding from private citizens too, endowments etc. People are "patrons" of the arts. Rich people supporting museums which will have some days that are open to all but charge admission normally. However, Washington DC is deliberately the peoples city. There's much for everyone To do there, even the poor. Though it's extremely expensive to live or rent there.
Both wasoxygen and flagamuffin made similar points, and after doing some reading on my own I must say that Polish museums do indeed get both private investors and governmental funding to some extent. I mean, most stuff outside of Louvre is unlikely to get by on tickets alone, and even that is my ignorant gut-feeling. Something just felt alien when I heard that I don't get a choice to go with a crowd for free or pay 4-8 USD (converted for you guys, depends on the museum) to get in. You can halve it if you are a student in all cases. I guess that all countries have their quirks, some are just less expected than others.