Is it possible to earn BTC by processing transactions? Where does the money go? Is it possible for any person to setup his wallet application to process transactions and make some more BTC?
If you run an Escrow service, yes, you could get paid to process transactions. But that's probably not what you're asking about.

In normal cases, transaction fees go into the reward for a new block being mined and are earned by the miners that create the new block. This takes specialised hardware and software. If you want to get your feet wet in this area, you ought to look into mining at a pool.