A private equity fund is an entity - usually a limited partnership - that is created by a “sponsor” (called a “manco” in that article for “management company”) and raises money from outside investors. The sponsor manages the fund and the investments it makes using that raised capital (and in exchange for various management fees paid by the fund to the sponsor). Note that, for a variety of reasons, those investments usually take the form of an investor (limited partner) agreeing to contribute cash up to a designated amount - but not upfront. Instead, the investor funds that agreement to invest cash in the fund when the sponsor asks for a portion of it to be sent to the fund so the fund can make an investment.
It’s been standard practice for a while for a private equity fund to borrow money secured by its assets, which are generally a combination of the ownership interests in the companies it has invested in and its right to cause its limited partners to contribute additional capital to the partnership. By using proceeds funded from debt, the fund can improve the way that its returns to its equity investors are measured, which results in the sponsor receiving enhanced incentive economics.
The main new development reported by this article is that private equity sponsors are now pursuing debt at the sponsor level that is secured by the sponsor’s incentive economics mentioned above as well as any management fees paid to the sponsors by the funds that it manages.
This doesn’t mean that the companies that private equity invests in are more indebted or more likely to collapse due to leverage - instead, the article is reporting that the overall structure of a private equity investment (an operating company owned by a fund that is managed by a sponsor) is seeing additional levels of debt at the top level of that corporate structure.