That's a bit of a culture shock to me that it seems like an exceptional thing that museum has free admission. Is it like that in USA as a whole? I don't know if that's a law or just a custom here, but all museums (aside of very short expositions or severely underfunded in my experience) provide access for free at least one day per week. I don't know if that comes from noble "anyone, no matter what is their financial state, deserves free access to knowledge and culture" or some weird law that allows, for example, lucrative tax break… but it's there. Warsaw also has "Night of the Museums" twice a year where all of them are open whole night and anyone can visit for free. It's a surprisingly huge event to be honest.
James Smithson left a substantial estate to his nephew, but when the nephew died without heirs Smithson's will stipulated that the funds be sent "to the United States of America, to found at Washington, under the name of the Smithsonian Institution, an Establishment for the increase & diffusion of knowledge among men." On August 29, 1838, eleven boxes of gold sovereigns arrived in New York, eventually funding the creation of today's collection of museums, which now receive about 60% of their budget from national taxes. It wasn't until I started traveling that I realized it was normal for museums to charge for admission. There are many private galleries in D.C. as well, but it can be hard to compete for visitors.
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.