Well theres a few ways to do it, ranging in difficulty:
The most basic way is to run bitcoind, a headless wallet, on a linux server, this is what I think they are doing, since their website went up in a time when bitcoin was less advanced than now in terms of online services. This method is as safe as your server is, I believe there was a scandal on lithium host where one of the employees simply opened up a customers vps and took the wallet.dat and thus were able to steal coins for themselves.

Another easier way, is to just use an online service, like bitpay, blockchain, or coinbase, and just listen for callbacks on a url you specified. This is easier and was less worries, as coinbase, bitpay, or blockchain take lots of precaution to prevent stolen coins and would possibly spot with their own money a theft (unlikely if large amount). These services usually have nicely generated html buttons to put on your website and actual payment pages you can setup.

An interesting service is inputs.io, they sorta function like visa in which you can pay/get paid with bitcoins, but the transaction happens off the blockchain, and instantly. Off the blockchain in the sense that when you pay with a credit card, actual physical money isn't being moved.The also provide a payment button html code generator and pretty easy api to work with.

Bitcoind, you are going to definately need knowledge of how to program. Online services, less so depending on what you will be doing, running a poker site you will 100% need to put together something in a programming lanuage.