How exactly does spacetime change inside a black hole?
You fall right across the event horizon without even knowing it is there unless you are paying attention.
In classical General Relativity, spacetime at the event horizon is locally Minskowskian, just like it is everywhere else in the universe where there is no curvature singularity. Thinking that something locally-bizarre happens at the event horizon — like space and time ”swapping”, whatever that means — is like thinking something bizarre happens at the North Pole. The Earth is just as locally-flat there, to first order in Riemann normal coordinates, as it is everywhere else.
Any “swapping” of anything at the event horizon is an artifact of a particular coordinate system, with no physical significance. Coordinates are assigned by humans, not by nature.
Inside and outside the horizon, there are three spatial dimensions and one temporal dimension, and you fall on a smooth geodesic through them until you reach the singularity.
As for whether quantum mechanics modifies this picture in a dramatic way (horizon firewall, anyone?), I don’t know, but I doubt it.
As for people who point out that outside observers don’t see you fall through the horizon, they're right, and they’re missing the point. What they see is completely irrelevant to what you experience.
A basic way to put this is that the points of a fall can define the distance. If you fall through a cloud then you might fall for 3 minutes along a distance of 60,000 feet with the underside of the cloud being point A and the ground being point B. Point A remains where it is. Therefore there is a spatial distance. Once inside the Event Horizon everything is falling together so you don't fall along a static distance anymore. You cannot define a distance because space itself is moving. Point A (space and anything else) inside the horizon is falling at the same rate as yourself. Normally we could plot your fall in terms of feet per second but inside the Event Horizon we could only plot your fall in terms of seconds because the place you fell from is falling too. Therefore even though you fell for three minutes it would only be in time.
As can be seen from the Schwarzschild metric for an r-coordinate r < 2M the term (1-2M/r) becomes negativ. This means that r- and t-coordinate interchange which does not mean that space and time interchange. This has two consequences:
The spacetime within the horizon is not stationary because r has the character of time, r decreases inevitably. The r-coordinate of a photon emitted upwards decreases. Hovering at constant r is not possible (time doesn't stand still).
r = 0 means that the singularity is a point in time (not in space).