Assume you want to prove a parabolic strict maximum principle, that is: Given an initial datum that attains everywhere a value $\ge 0$ but is not identically zero the solution of the corresponding linear heat equation is $>0$ everywhere and for all time $t>0$.

One possibility is the following: One shows that (because the semigroup that yields the solution is analytic) the above property is equivalent to irreducibility of the semigroup.

Now, irreducibility of the semigroup is (at least in my eyes) a very physical property of a diffusion process: It essentially says that you cannot expect particles to consistently hit a certain point and being reflected if that very point is not part of the boundary at all (say no potential-generated barriers).

(Indeed you can formalize this latter reasoning, but in my opinion even the introduction of the notion of irreducibility in the semigroup theory is perfectly justified by physical reasons.)