How does a stock buyback work in this case - do they just buy up 51%? If so do the remaining 49% of shareholders get value 'extracted' on their behalf too or do they also get screwed?

Who is lending money to a business so the business can pay its owner?

Large companies can just sell debt on the open market. When Bain's subsidiary "merged" with Guitar Center, Guitar Center sold $1.78 billion in debt and Bain paid $1.9bn for the company. Guitar Center got murdered on the interest rates of between 6 and 10% on these debts, which were rated as junk even at the time. The details are in the various SEC filings, you can read them online.

"""In connection with the execution and delivery of the merger agreement, Parent and Merger Sub entered into a debt financing commitment letter with J.P. Morgan Securities Inc. and JPMorgan Chase Bank, N.A. to provide up to $1.815 billion in debt financing, consisting of (1) a senior secured asset-based revolving facility with a maximum availability of $375 million, (2) a senior secured loan term facility in an aggregate principal amount of $800 million, (3) a senior unsecured bridge loan facility in an aggregate principal amount of up to $300 million and (4) a senior subordinated unsecured bridge loan facility in an aggregate principal amount of up to $340 million. """

As to the question of why company management would agree to do this, it's because they personal get paid a huge amount of money to execute the merger. Guitar Center's old CEO got paid $25 million by immediately paying out all his future stock grants on the date of the merger.

"""Each stock option to purchase our common stock outstanding immediately prior to the merger, whether or not then exercisable or vested, will become fully vested and will be deemed to be exercised and canceled at the effective time of the merger, and each holder of such stock option will be entitled to receive a cash payment..."""