Well, it does cost money to recycle things, but the key is that it often costs less to recycle something what the end product is worth. As a result, even though there is cost involved, people still make a profit.

As an example, there are recyclers in every city that will pay you money for metal you are willing to sell them. Do they do this to be charitable? Nope, they're in business to make a profit. Extracting the ore from which metals are derived, then purifying the metal from the ore is a very expensive process. Recyclers pay to get scrap metal because much of the work is done for them, and they can reprocess the scrap metal at a low enough cost to sell it at market value and turn a profit.

Another example: in my city, the Lions Club has dumpsters in many grocery store parking lots for people to drop off newspapers. Why? Because they can sell the newspapers to a recycler. The recycler is willing to pay for the newsprint because they can reprocess and sell at a profit. Again, much of the work is done for them already (pulping wood, etc), so they can buy old newprint, reprocess it, and sell it for a profit at the going rate.

So yes, there can be significant costs to recycling, but there are also significant costs to producing something from scratch; the cost to make something from scratch often outweighs the cost to recycle it, so that's where the $$$ comes from.