Why does finer mesh mean worse condition number? Suppose I am working on the finite element approximation of a problem. My understanding is that the condition number of the resulting algebraic system becomes worse when the mesh becomes finer. What is the reason for this?

It is not universally true that the condition number becomes worse with a finer mesh, for example take $10u = f$. There's no good reason to solve this via FEM, but you can do it, and the condition number doesn't increase with mesh refinement. Less trivially, if the operator in question is bounded and strictly coercive, then the condition number of the operator is finite and as our mesh refines the condition number of the matrix should converge to the condition number of the operator.
The problem is that bounded and strictly coercive operators rarely model interesting physics. Hence the condition number of the operator is unbounded and as we create better approximations to it, the condition number of the approximating operator becomes worse.
