Let's say your friend is selling a beat-up old car for $500.
You think the car's got some potential, so you go to the bank and borrow $400, and you pony up $100 of your own money to buy it. You've now just entered into an investment funded by 20% equity and 80% debt.
Now you start fixing up the car in your garage to increase its value. You fix the engine so that it quits making that popping noise, and you apply a nice new shiny coat of paint to the car. All this costs you an extra $100. On the side, you also make $100/month giving your friends rides to work every morning using the car.
You use that $100/month to start paying down your original $400 loan from the bank (+ whatever interest you've accumulated), and after 5 months, you pay it down completely. You've just deleveraged your investment. What's more, your car is now worth $1000since you fixed it up and it doesn't make that popping sound anymore. So you exit your investment by selling it to a used-car dealership.
So what's the damage? Well, you really only had to invest $200 (your original $100 + $100 to fix it up), and you ended up selling it to the used-car dealership for $1000. You've made 5x your money in 5 months! Not too shabby!
Private equity is basically the same thing, though with real companies and over larger time scales. They take cheap and underperforming companies with potential, take them private through a leveraged buyout (LBO), and eventually exit at a profit by selling it to someone else.