Typically, PE companies buy the company while only putting down a small portion of the purchasing price. They finance the rest through banks. The whole concept is pretty similar to buying a house with a mortgage, except you're buying a company. The debt payments are then a tax write-off, and all/most available company cashflow is diverted to paying off that loan. The ideal company is one with reliable cashflows and access to a growing market (of which I'd say a large cloud infrastructure provider fits the profile).
To improve cashflow: pay fewer people less, convert assets to cash, and/or make more money from your core business. Usually a mix of the three.
From an organizational standpoint, you can think of it like Rackspace just took out a very large (maybe subprime) mortgage on the company and some new people are going to try to make sure that doesn't turn out to be a horrible bet, first for the investors, then for the company.
