In this paper, the stability of a discretization $$L_hx_h=y~~~~~~~~~~~(*) $$on a continuous problem $$Lx=y~~~~~~~~~~~(**)$$ where $L\in BL(X,Y)$ and $h$ represents the mesh parameters, is given by the uniform boundedness of the inverse operator $L_h^{-1}$. That is, the numerical scheme $(*)$ is said to be stable if $$\lVert L_h^{-1} \lVert \leq C,$$ for some $C>0$ irrespective of the family of mesh parameters represented by $h$.
My doubts:
How can we ensure the boundedness of the most often partial differential operators appear in $(*)$ as we have seen that derivative operators are not continuous?
How can we practically explain the connection between stability and the 'uniform boundedness'?
One notion of stability is that "for small differences in input, we get small differences in output". Mathematically, it could be, e.g., for two pieces of data $y_1, y_2$ and their respective solutions $x_1,x_2$, you would want $$\|x_1-x_2\| \leq C \|y_1-y_2\|.$$
If $y_1$ and $y_2$ are 'close' in some sense, then the solutions to the inverse problem should also be 'close', perhaps up to an extra multiplicative factor $C$. The uniform boundedness of $L_h^{-1}$ gives precisely this notion of stability, since (for this system) a solution is produced via $x=L_h^{-1}y$. So
$$\|x_1-x_2\|=\|L_h^{-1}y_1-L_h^{-1}y_2\|\leq\|L_h^{-1}\|\|y_1-y_2\|\leq C\|y_1-y_2\|.$$
Why is $L_h$ bounded?: Since (as far as I can tell) $L_h$ is acting on a finite dimensional discretization. Even though it represents differentiation, it is still bounded since it is acting on a finite dimensional space (and all linear operators in finite dimensional spaces are bounded).
Concerning $(**)$: this is a very good observation, and in general, a linear operator involving differentiation will not be bounded in an infinite-dimensional space (and its inverse may not even exist)! So very careful analysis must be done in order to ensure that, as the grid $h$ refines, you actually converge to a solution of the continuous problem. Check out the literature/proofs on Galerkin methods for more info on this :-)