The answer is actually quite simple.
Any aerodynamic body, like a wing, generates a lift and a drag, which are normally represented as coefficients vs. angle of attack. A classical airfoil like a NACA 2412, used for example for the wing of the Cessna 172, has the following $C_l$ (in blue) and $C_d$ (in red):
Now, lift and drag are not the end of the story. To completely describe the aerodynamic characteristics of an aerodynamic body a third plot is needed which is as important as the other two but which is normally ignored: the plot of the pitching moment. For the same NACA 2412 it looks something like that (in green):
What does this plot tell us? The pitching moment is normally (far from stall) negative, that is, the pitching moment is normally nose-down. And that's why the horizontal stabiliser normally produces a downward lift. What happen at stall? As soon as the stall region is reached it becomes suddenly even more negative, almost five time more negative than just before the stall! What does imply all that at stall?
- Lift drops as well as the airplane;
- drag rises, slowing the airplane even more down;
- and nose-down pitching moment increases, making the nose of the airplane going down.