I've worked on a site that does something similar - different, but similar. The site I worked on allowed people to create conference phone calls over the internet. I built an admin interface that allowed people to do things like creating the call itself (this would generate the room number and display it on the website - you could then call the number up from any phone and join the room by entering the code displayed on the site) and allowing people to download mp3 recordings of the conference calls.
I can tell you we achieved this most likely in much the same way that this site works - the php code just calls an external api that does the phone related stuff. In my case, I was physically sat next to the engineer who was working on the phone software (internally, he was using a program called Asterisk in a custom way to do what we wanted to do: http://www.asterisk.org/) - and we worked on the integration together. In the case of the site you have posted - essentially the website code will be calling some other service somewhere - this could be something they've done internally, or some external api that can be used for this effect.
So basically, the php code calls an api that does the phone related stuff.