How? Per the blog post: "OpenAI’s board of directors consists of OpenAI chief scientist Ilya Sutskever, independent directors Quora CEO Adam D’Angelo, technology entrepreneur Tasha McCauley, and Georgetown Center for Security and Emerging Technology’s Helen Toner." That's 4 directors after the steps taken today. Sam Altman and Greg Brockman both left the board as a result of the action. That means there were 6 directors previously. That means a majority of 4 directors. Assuming Sam & Greg voted against being pushed out, Ilya would have needed to vote with the other directors for the vote to succeed.
Edit: It occurs to me that possibly only the independent directors were permitted to vote on this. It's also possible Ilya recused himself, although the consequences of that would be obvious. Unfortunately I can't find the governing documents of OpenAI, Inc. anywhere to assess what is required.