Nobody knows yet why the board relented, but the scuttlebutt seems to be that Twitter arranged an internal valuation once Musk put a price on the company, presumably as justification for an internal plan that they'd announce as their alternative to acquisition, and the valuation actually showed Musk was overpaying for the income Twitter was likely to generate.

(If the board hadn't relented, Musk's next step would have been to arrange for a tender offer, which goes directly to the shareholders, but does not involved Musk acquiring new shares, but rather just a commitment to buy those shares if the board drops the poison pill. Successfully getting that commitment would be a very strong signal to the board, which would likely then drop the shareholder rights plan and allow the acquisition to proceed. But Twitter's board skipped those steps.)