What's the physical meanning and principle of applying anisotropic diffusion in transform domain？
Physically, there is usually not much implied meaning (but this may depend on a case-by-case basis). Transforms like shearlets are actually used to embed the data into anisotropic or edge-aware transform domains.

Mathematically, it may make more sense. Isotropic diffusion tends to create blur (smear) in whatever domain it is applied. Thus, if you want to obtain a result that is compact in the considered domain, be it transformed or not, applying non-isotropic diffusion is very tempting.

Another way to see it mathematically, is to understand that isotropic diffusion corresponds to solving a least square problem that will create many small non-zero coefficients by trying to spread the error everywhere. However, a helpful and widely available assumption about signal is their sparsity in the transform domain, i.e., they have few non-zero coefficients in a suitably chosen transform domain that concentrate the energy. This corresponds to solving problems where the exponent of the norm of the prior term is less than 2 (typically 1) and thus the use of anisotropic diffusion to find a solution.