What it actually was
The Bitcoin Foundation was a corporation with members, a board and a published rulebook. Its bylaws, kept in a public GitHub repository so that changes arrived as pull requests, open with the line "BYLAWS of THE BITCOIN FOUNDATION, INC. (a District of Columbia non-profit corporation)" and are dated effective 23 July 2012. Article II states the purpose as an association of persons with a common business interest operating under section 501(c)(6) of the Internal Revenue Code, which is the trade association part of the American tax code rather than the charity part.
Article III created three membership classes: Founding, Industry and Individual. The founding members are named in the document, and one of them is listed as "Satoshi Nakamoto, at [email protected]", identified by the whitepaper and a PGP fingerprint and by nothing else. The governance page set the board at five seats, two elected by individual members, two by corporate members and one by the founders, on two-year terms.
What it did
It paid people to work on Bitcoin Core, and it turned up in Washington. On 18 November 2013 its general counsel Patrick Murck testified before the Senate Committee on Homeland Security and Governmental Affairs at a hearing titled "Beyond Silk Road: Potential Risks, Threats, and Promises of Virtual Currencies", describing the Foundation as "a member-driven nonprofit" and telling the committee that "there is no Bitcoin company that manages or controls the software or its operation". In 2013 almost nobody else in Bitcoin had a lawyer on retainer or an address a regulator could write to.
How it ended
It ran out of money. The Internal Revenue Service filings reproduced by ProPublica show 2014 revenue of $915,443 against expenses of $3,285,602, a shortfall of $2,370,159, followed by $395,538 of revenue in 2015. The developer payroll went elsewhere: MIT's Digital Currency Initiative says it "was formed in 2015 in order to provide a stable and sustainable funding for long-term Bitcoin Core developers". The Foundation kept publishing board minutes into late 2016 and described itself in 2016 as focused on education, advocacy and adoption. Its last recorded filing is for 2018, and ProPublica notes it "is not listed in the IRS's most recent list of tax exempt organizations". The bitcoinfoundation.org domain now serves an unrelated cryptocurrency news site.
Nothing was seized and nothing was wound up in court. It simply stopped.