In this diagram, as well as our universe, you have a parallel universe. Where does this come from? Is this just a artifact of the diagram, or is it predicted by the maths in some sort.
This is the 'maximal extension' of the Schwarzchild metric - the solution of the field equations for a point mass that exists eternally, both in the future and in the past. It doesn't describe our universe, or real astrophysical black holes, because they don't extend forever into the past. (Our universe started with the big bang, and black holes started with a collapsing star.)
Schwarzchild's original black hole solution only described the right-hand and top quadrants. You have the outside universe, and the inside of the black hole that things fall into. If you start with a freely-falling particle with a given position and velocity, you can trace its trajectory forwards in time and see it fall into the black hole.
But you can also trace the trajectories backwards in time, too. The equations are time-symmetric, so any solution can be time-reversed and yield another equally valid solution. However, if we trace the trajectories backwards in Schwarzchild's right-upper diagram, we run into the diagonal line on the edge of the diagram, where it stops. What then?
Likewise, for locations inside the horizon, for a given velocity you can trace the free-fall trajectory back in time to this 'edge'. Spacetime is smooth there, there doesn't seem to be anything special or extreme going on there (unlike the singularity at $r=0$). So what happens? Is there really an 'edge' to space and time there? Or is it just a coordinate-system thing, and spacetime carries on?
The maximal extension is what we get if we assume spacetime carries on. It's like the way Eratosthenes observed that the Earth was curved in the neighbourhood around Egypt, and extended that to work out the circumference of the world. We use the fact that General Relativity is time-symmetric, turn the diagram upside-down, and glue the two halves together. The result is a full solution to the equations of General Relativity that is time-symmetric - it extends forever both forwards and backwards, and looks the same both forwards and with time in reverse. All geodesics (freefall trajectories) end either at singularities or infinity.
It answers the previously-awkward questions about why the diagram is not time-symmetric, and what you would see if you sat at a point and looked in that direction. (We can only see events along the past light cone, pointing downwards on the diagram. We have to trace trajectories of light rays back in time.)
Another useful way of looking at the diagram is to consider the analogy between black holes and uniform acceleration. Rindler noted that the diagram for a maximally extended Schwarzchild black hole looked a lot like the diagram for a uniformly accelerating reference frame. (This is to be expected by the equivalence principle.)
(I should emphasise, they're not actually the same metric, just similar-looking.)
A uniformly accelerating particle follows a hyperbolic trajectory when drawn on a spacetime diagram. This looks very much like the hyperbolic trajectories of particles hovering at a constant radius outside a black hole. Its past and future approach two light rays. The accelerating particle can never see any event that occurs after the futurewards light ray - it acts like an event horizon to the accelerating observer. If it ever crosses the line, it can never escape back to its home quadrant. And similarly, it can never influence any event before the pastwards light-ray - it acts like the anti-horizon in the eternal black hole.
So, we can take any patch of empty space, draw the spactime diagram for it, and draw an analogy with a black hole using accelerated observers. A uniformly accelerating observer constrained to stay within the right-hand quadrant sees the two light rays emanating from the centre as event horizons. They can see events in the past light cone, the bottom quadrant. And they can send messages to observers in the future light cone, the upper quadrant. But they can neither see nor be seen by observers in the left-hand quadrant. It is forever 'elsewhere'.
The accelerating observer can deduce it's there by tracing trajectories in the top quadrant backwards, or the bottom quadrant forwards. But no communication is possible. You can't get there without going faster than light, and they can't get to you.
So, a uniformly accelerating observer could deduce the existence of a part of the universe they can never see or influence, by considering what free-fall observers beyond their event horizons could see or be seen by. You just take the part of the universe you can see, and extend it using the same laws of physics as far as it will go.
Because the problem seems to be trying to visualise how this geometry connects together when we add more dimensions, I'm going to attempt to draw some pictures!
First, get two sheets of paper and sketch on them both something like the following:
This shows the lightcones pointing out of the singularity in the bottom half, crossing the event horizon and progressing up the sheet, and then diving back in to the singularity. Fold the sheet at right angles along the event horizon.
Next make a cut in each sheet from the singularity to the event horizon, and tape them together in the following shape.
Note that each sheet joins on to one sigularity in the top half, and the other singularity in the bottom half.
Now, if you take this model and give one of the sheets a twist through $180^\circ$, you get the Penrose diagram you showed above. Try to flex it so all the lightcones point up. (You really do have to make one out of paper and try it to get the idea!) This is effectively taking a copy of the Schwarzchild solution, time-reversing it, and gluing the two together, as mentioned above.
Finally, to appreciate what happens in 2+1D, we take one of the sheets and rotate it about the singularity like this:
The top half of each sheet spins around one $r=0$ singularity, which we can imagine being the equatorial plane of the black hole in our universe. The bottom half of each sheet connects to the singularity from the other time-reversed universe.
And then you have to imagine adding the other sheet, spun around its own singularity, joining up continuously.
Thus there are two whole external universes, joined together inside the black hole.
That's a pretty crude way of visualising the geometry, and we're still missing a dimension! It's hard to cram multisheet 4D spacetime wormhole geometry into our 3D imaginations, which is why we generally stick to the simple, flat Penrose diagram.
I doubt that helps much, but it's the best I can do right now.