Private equity firms have a unique structure: The general partners (GP's) of the firm create funds and manage the investments of those funds. Limited partners (LP's) contribute the capital to the funds, pay fees to the GP's, and then make money when the funds' assets grow.
I believe the article is saying that ultra high net worth individuals participate in the real estate market by hiring someone to act as a general partner and manage the real estate assets. They and their friends contribute the cash and get shares in the resulting fund.
Usually this GP/LP structure is used when the funds purchase or invest in private companies, which is why it is referred to as "private equity structure," but the same structure can be used to purchase and manage pools of real estate or any other investment asset.