Same as if they were taking a public company private. They paid the current owners for the rights to the company.

Issuing equity to employees in the new company is a different concept.  They would issue equity in the company to incent the employees to act in a manner that is aligned with their goals to grow the company.  As equity holders (ie owners in the new company) they would receive a payout when the PE company sells the company to a new owner at some point in the futures.