If you plot a continuous curve in spacetime, it could be a path of a body if the tangent to the curve exists and has positive squared interval.
General relativity is a geometrical theory, so everything is written in a geometrical way and the geometrical generalization is the predictions the theory makes.
So outside the event horizon your curve has to have t change more than the others, so in particular t has to change and get larger.
Inside the event horizon your curve has to have r change more than the others, so in particular r has to change and get smaller.
The Schwarzschild solution isn't clear about the horizon itself because the coordinates themselves break down there. However there other coordinate systems that are not weird there that make the same predictions inside and outside the horizon and you can track where thing that are outside go when they go through. If you do, the curves of decreasing r on the outside become curves of decreasing r on the inside.
So that is the reason that when your curve has decreasing r on the outside it starts out with decreasing r on the inside.
As for why it stays decreasing. We have to have r change (because it has to change more than the other coordinates to have a positive tangent). And since there is one + and three - there isn't room for a changing and decreasing r to turn around into a changing and increasing r.
The same thing happens with time outside the horizon there is a cone of time increasing more than space changes and a cone of time decreasing more than space changes but they only interest where space doesn't change at all. If you insist that time changes enough more than space that you get a positive tangent then you have a hyperboloid of time increasing and a hyperboloid of time decreasing and they don't intersection. So for positive unit tangents there is no way to get from one to the other on a continuous way.
Outside this means things don't start going backwards in time so it doesn't seem weird. Inside this means that if r is decreasing then it can't change to start increasing or even staying the same.
So the same reasons you have to go to the future when outside make it so that you can't increase your r when you are inside.
The equations for the inside might look less mysterious if you wrote r as t and vice versa, but they would be more complicated. Since there is a symmetry in t where the metric doesn't depend on t. So since the metric doesn't depend on t it gets a nice simple form when everything is written in terms of r. However r is your time on the inside so the metric will start crushing dynamically the
$r^2 d \Omega^2$ now just makes an angular coordinate separation becomes closer and closer dynamically as the incoming curve extends. The t coordinate represents a different direction you can turn around in but all three spatial coordinates don't affect the metric, so the metric changes dynamically and there is nothing you can do about it.
The metric changes and no matter how you fire your rockets it changes and the tidal forces get stronger and stronger. You still have a 3d space to move around in it's just that no direction gets you to your past and the metric is no longer a function of how you turn around in space and it just gets stronger and stronger.