The Schwarzschild metric is relatively easy to visualize even though the metric is singular at r=2M and r=0. Once we make the Kruskal extension, we find whole new regions in the manifold including regions we call White Hole and if we take a t=constant slice we get an Einstein-Rosen Bridge too.
Mathematically, this is how I think about the Kruskal extension. (Please correct me if I am wrong) To get a solution from the Einstein's equations we can think about a metric which has some particular symmetry. If we have a spherically symmetric system, then we argue the form for the metric (the coordinates in which that symmetry is most obvious are called the Schwarzschild coordinates.) and resubstitute it in the Einstein Field equations to find the form of the components of the metric.
Now, we found that if we change the coordinate system from Schwarzschild to Kruskal we get new regions and the reason for that is that the Schwarzschild metric didn't cover the whole manifold and because Kruskal is the maximal extension we know the Kruskal coordinates cover the whole manifold. But, on the other hand, if we had somehow started with the form of the metric which looks like the Schwarzschild metric in Kruskal coordinates, we could have got an analytic solution in a coordinate system which covers the whole manifold in one go.
But, is there a way to gain physical intuition over how the whole blackhole (i.e in Kruskal coordinates) looks in reality? I know I am being vague here but, I am not able to explain it better. If we take a constant time slice we find a geometry which we call the Einstein-Rosen bridge. So, does a Schwarzschild Black Hole which forms in our universe really have that geometry.
Or, is it that once we find an analytical solution to Einstein's Equations and find the maximal extension we have to rule out certain region through physical intuition?