How websites like silkroad and pandora process transactions?
The underlying mechanism is to generate an address specifically for one payment. The address space for Bitcoin is mind-boggingly large (almost any 256 bits number is an address) so that's not an issue. That address is monitored for a payment and once the payment is received, automatically transferred to the merchant's final address (or payment).

As to software, it's a bit all over the map. You can either use something that will make the whole process transparent (like BitPay or Coinbase) or use lower-level APIs interacting with the BlockChain like BlockCypher.