Picture below is from 260th page of G. Huisken's Flow by mean curvature of convex surfaces into spheres.
First, I think it should be $$\widetilde h_{ij}(x,\widetilde t)=\psi(\widetilde t) h_{ij}(x,\widetilde t).$$
Second, I don't know how to differentiate (14). Consider locally, then $$ \int_{\widetilde U_t} d\widetilde\mu=\int_{\widetilde F(U,t)}\sqrt {\widetilde g(x,t)} dS=\int_{\psi(t) F(U,t)} \psi(t)\sqrt{g(x,t)} dS $$ Then I am not sure how to proceed.
Just one way of all, and there should be some way is more easy. In fact , the area of $M_t$ is $$ \int_U \sqrt {\widetilde g} dx =\int _U\psi\sqrt g dx $$ Then $$ 0=\partial_t \int _U\psi\sqrt g dx =\psi '\int\sqrt g+ \psi\int -\frac{1}{n}H^2\sqrt g $$ So $$ \frac{\psi'}{\psi}=\frac{1}{n}\frac{\int H^2\sqrt g}{\int\sqrt g} $$