Free classified website normally has thousands of (free) pages and all these pages are platforms for advertisement. Majority of the revenue of free classified websites comes from the ads being displayed on the page.
When someone visits the page of these free classified websites, ads displayed by different publishers (like Google AdSense, InfoLinks) are clicked or viewed by the visitors. Thus free classified website makes money.
Sites like Quickr and OLX operate on Freemium model. They offer their basic services for free, if some seller wants to get more coverage - he can opt for their premium services. Quickr earns through advertising on website. A majority of them depend on the advertising revenue like Google AdSense.
As per my experience I have known that there is much contribution of the advertisement on any website for earning good and here is the same case that the free classified websites make money through the advertisements displayed on their webpages. Like if any user clicks on those ads and complete the required action then the sites are paid by Google for that. So the free classified makes money through the number of ads on their various webpages.
I find that the harder I work, the more luck I seem to have.
Advertising that stays on 1000 impressions or per month is one way to make money. Another way is to get paid placements after the free placement days are over. Some of the placements site also collect the email and phone number of the people who make posts. This way the data collection from such sites is being sold to the marketing companies who need email or mobile for their own marketing purpose. It does not make much money but it is possible to make some money.