Here is how private equity works in a nutshell. There is an entrepreneur out there with an idea for a business. He creates a new company and starts his business. He is the sole proprietor in the business and gets things going. He quickly finds out that in order to keep his business running he needs money and lots of it. He doesn't have it. In order to get the money needed, he has to ask for it (see the show Shark Tank.)
He pitches his business to different investors. Eventually he finds one willing to give him the $500,000 or whatever he needs in exchange for 40% of the company. He agrees. With the funds he is able to buy the equipment he needs to fulfill orders. He starts making sales, then more and more sales. Soon the money coming from the sales is enough to hire workers to help him fulfill the orders. Soon the revenues generated from the sales minus the expenses leave a little profit. That profit is either reinvested in the company, like buying new equipment, maybe a bigger building, a new website, etc. or is distirbuted to the owners, 60% to the entrepreneur and 40% to the investor.
Perhaps after 2 years of this the entrepreneur decides he wants his whole company back. He can buy that 40% back from the investor, probably for a lot more than the original $500,000, and have complete ownership again. The investor made a profit, which was the goal from the investment, and the entrepreneur now has a profitable business up and running thanks to the initial investment to get things off the ground.
Your question asks for examples of this. I think you would be hard pressed to find any company or business that didn't go through a cycle of some kind just like this. It is incredibly common. Almost every company you have ever heard of needed money to get off the ground. That money came from an investor 9 times out of 10, whether it was a private equity group, a bank, whatever. A business typically needs money to get started.
In regards to Romney and Bain Capital, they funded dozens of businesses. They also founded dozens more. They also acquired dozens more on top of that. Whether Bain helped the company, took over the company or started the company, the goal was always the same: make the business profitable. Then they could sell their share for a big return or keep it as an asset. Almost every single company Bain touched grew and improved. Many of those businesses now employ more people than they did when Bain invested. Some have since gone bankrupt. If you look at all the jobs lost by companies Bain helped and all the jobs gained since Bain helped them, the net job change is positive.