He can borrow against his Tesla and Space X holdings. The Twitter stake itself can be used as collateral. Or he can sell shares of them.
There is LBO financing, where the purchased company itself borrows to fund the purchase. But I doubt a company as risky as Twitter can borrow a significant fraction of the purchase price. And Musk doesn't have a history of using financial engineering like that.
You're mixing terms. "Leveraged buyout" is really a term of art referring to a purchase largely financed by debt issued by the purchased company. It only works with well established businesses with really reliable revenue streams. If the business fails, the debts go unpaid.
I would be surprised if Twitter could raise as much as 10% or 20% of its own purchase price.
if the board wanted to sell they would establish that in a reply and ask for the details of his financing. They don't want to sell and financing isn't the issue, its just a red herring.