To finance the purchase, the firm arranges debt with a lender and sells shares in the LLC to raise the equity needed to cover the difference between the purchase price and debt.
In a typical transaction, a private equity firm finds a property they would like to purchase and they form a separate legal entity, an LLC, to do so.  To finance the purchase, the firm arranges debt with a lender and sells shares in the LLC to raise the equity needed to cover the difference between the purchase price and debt.