First, rather than viewing the mean curvature as "the trace of the second fundamental form," it might be more intuitive to view it as the average of the principle curvatures, i.e., $$H = \frac{1}{n} \sum_{i=1}^n \kappa_i \;.$$
Second, it may be that the following view of minimal surfaces (those of mean curvature zero) from D. Hoffman and W. H. Meeks III, in their paper, "Minimal surfaces based on the catenoid" [Amer. Math. Monthly 97(8) (1990), 702-730] (ACM link), might help:
“Loosely speaking, one imagines the surface as made up of very many rubber bands, stretched out in all directions; on a minimal surface the forces due to the rubber bands balance out, and the surface does not need to move to reduce tension.”
You can see this accords with Peter Michor's more abstract formulation.