Greene's book contains this picture (figure 3.8b):
It's schematic (these discrete squares don't really exist), but it's meant to illustrate how the universe can be finite with respect to one notion of time and infinite with respect to another. If you take the "current time" to be the vertical axis, then the inflationary bubble only spans a finite number of squares at each time, and it has a boundary. If you take the "current time" to be the value written in each square (which is supposed to represent the energy density of either the inflaton field or the matter that it decays into), then at any particular time other than 100, only the interior of the inflationary bubble exists, and the rest of the universe is in the past. Both of these time coordinates respect causality (the past light cone at any point only contains points at later times, for either notion of time), so they are both equally valid.
The value-in-the-square time coordinate makes sense if you want to talk about events inside the bubble, because all of the squares with the same value in them are essentially interchangeable: the universe at any particular value of that time coordinate is homogeneous and isotropic. (It looks as though the square at the vertex of the V is special, but that's just because this model with squares is not very accurate.) The value-in-the-square time is what's called "cosmological time" in the standard cosmological model.
If you want to talk about events outside the bubble, then the value-in-the-square time is useless since it is ill-behaved (degenerate/singular) in the region where all squares say 100. The vertical-axis time doesn't have that problem, but it is less useful for events inside the bubble since it doesn't respect the symmetry of the bubble.
You could say, as Greene does, that the value-in-the-square time is the time for an inside observer and the vertical-axis time is the time for an outside observer, but I dislike that language, since it suggests that you are somehow obliged to choose particular coordinates based on your location. The reality is that you can use any coordinate system that covers the region of the universe that you're interested in. Some coordinates are more convenient than others, but none is ever wrong.
Also, Greene says
Trixie, floating [in the 100 region], is observing the formation of a nearby bubble universe. Focusing her inflaton-meter on the growing bubble, she is able to directly track its changing inflaton field value.
which makes no sense: it's impossible to see the bubble from outside, since it expands at the speed of light, and no information about it can propagate faster than that. You can't even know that a bubble exists there, since it starts from an uncaused random fluctuation. The only way to measure those numbers is to wait for the bubble to engulf you, after which you can never leave. So really Trixie ought to be considered an inside observer, despite her choice of coordinates.