How did he effectively shut it down without explicit constitutional authority?
But the creation of such a bank is not in the Constitution; it was established by an act of Congress and signed into law by President Madison.
The charter for the bank was set to expire in 1836. With Jackson's known opposition to the Bank and with the upcoming elections in 1832, Senator Henry Clay thought to corner Jackson politically by renewing the charter before the elections, according to Edward E. Baptist's The Half Has Never Been Told. The renewal overwhelmingly passed the Senate and then the House in the summer of 1832. However, on July 10th, Jackson announced his veto, and the Bank charter would expire.