The model they constructed has the three-dimensional universe floating as a membrane (or brane) in a "bulk universe" that has four dimensions. (Yes, this is making our heads hurt as well, so it might be easier to temporarily think of the brane as two-dimensional and the "bulk universe" as three-dimensional when trying to picture it.) You can read the more technical details in this 2000 paper on which the new theory is based.
So if this "bulk universe" has four-dimensional stars, these stars could go through the same life cycles as the three-dimensional ones we are familiar with. The most massive ones would explode as supernovae, shed their skin and have the innermost parts collapse as a black hole.
The 4-D black hole would have an "event horizon" just like the 3-D ones we are familiar with. The event horizon is the boundary between the inside and the outside of a black hole. There are a lot of theories of what goes on inside a black hole, although nothing has ever been observed.
In a 3-D universe, the event horizon appears as a two-dimensional surface. So in a 4-D universe, the event horizon would be a 3-D object called a hypersphere.
So basically, what the model says is when the 4-D star blows apart, the leftover material would create a 3-D brane surrounding a 3-D event horizon, and then expand.