Well these are not the best answers guys. I think the OP is asking how CPA companies make money, not how to make money with them. The most basic way CPA companies make money is through the relationship they have with their advertisers. For each action your end-user takes that the CPA company pays you for - their advertiser pays them MORE than what they pay you. In this way, CPA companies make their money through one of the oldest money making mechanisms - arbitrage.
They also do almost no work, other than brokering an indirect deal between you and the advertiser. Basically affiliates do most of the work by promoting the offers, advertisers pay the CPA network based on the lead volume, and then the CPA network pays you your share. That's how it works. Of course the CPA networks do have other revenue models as well, but this is the most basic way they make money, and the way they make the majority of their money.