How do private equity find private companies that fit their investment criteria?

There's no set rule but if the company is of any size it's likely through a banker. Other ways are through other professional service providers (lawyers, accountants, consultants) through PE guy's general network (a friend of yours knows you're in PE and they know a guy who knows a guy who wants to sell a business-one time we did a deal when I was younger where we found it because one of my senior guy's son played on the same Little League team with a guy's son who was a CEO who wanted to do an MBO), some firms are big into cold calling (popular in growth equity and companies that do platforms and a lot of smaller bolt-ons), there are databases like Capital IQ among others, you may end up reading about a company that's in the news for something unrelated and it sounds interesting so you do a little research and call them up, you tend to focus in industries so you know what's going on and the companies in those industries and hear that someone's interested in selling (or get an inbound call saying they're interested in selling), from other PE firms who may be exiting or who hear about a deal that's not in their wheelhouse but they know it's in your's so they'll refer you, and probably others I haven't listed. Generally you're not going to pre-analyze a potential acquisition of a private co because you won't have much or anything in the way of financials so you'll sign an NDA and take a look under the hood. And you tend to do an initial look on hundreds of possible deals for every one you do, or really any one that you go beyond the initial look at.

Like I said, there's no single method. The best deals are off market and not marketed by an IB because they're not run through an auction but they're far and few between today. I'm assuming you're not already in IB so most likely the interviewing firm wasn't looking for you to find them a company but they wanted to see your thought process.