No matter what, they can't go outside of what the DVD specs allow. My bet is that this is done as a combination of menus interweaved with regular video footage.
So you start playing the video, then just before you get to the inifinifilm content, you move to a motion menu that continues the video, then using delayed menu buttons, you bring in your extra choices - the user selects the choice and you branch off to the extra's clip. When that's done you come back to the menu at which point you choose another extra or continue forward with the movie.
You could do a simple one using DVDA3, or for that matter even DVDA2 - the only thing to keep in mind is that all of the menus cannot exceed 1GB of space. The 1GB limitation is where something like DVD Lab Pro or Scenarist would help, since you could then put multiple VTS's on the DVD, each with its own 1GB limit.
Now, the internet enabled content is probably done with something like Sonic's eDVD product (and since it requires the interactual player, it's a good bet they use eDVD or something similar).