Say you have some PDE for a function $f:[0,T]\times \mathbb{R}^n\to\mathbb C$
$$\frac{\partial f}{\partial t}=Lf,$$
for some differential operator $L$. If $L$ contains a Laplacian term $$\Delta f=\sum_{i=1}^n \frac{\partial^2 f}{\partial x_i^2},$$
why do people claim (as a rule of thumb) that this helps "smooth the solution"? From my understanding they are saying that the solution of the PDE will have better differentiability, but I am unsure why the Laplacian has this effect.
The Laplacian itself does not smooth the solution. What smooths the solution are the dynamics $${d f\over dt} = |\kappa| \nabla f + \text{other stuff}$$ The intuitive reason is exactly the opposite: the Laplacian amplifies regions with large curvature as compared to regions that are relatively flat. The time derivative "pushes" $f$ more in regions where $f$ is rough, and the signs are such that the pushing is towards a smoother $f(x,t)$. If instead the equation was $${d f\over dt} = -|\kappa| \nabla f + \text{stuff}$$ The Laplacian would cause large curvature regions to get even rougher.