How is the first variation formula applied here?

Regarding Lemma 9.3 (page 262) in this paper by Huisken, I have two questions.

1. It is said $$\vert \tilde{M}_{\tilde t} \vert = \frac{1}{n}\int \tilde H (\tilde F \tilde \nu) d\mu$$ follows from the 1st variation formula.

I can only recognize the integral is the variation of area functional, but I don't know how to derive this equation from the 1st variation formula.

1. Why the enclosed volume $$\tilde V$$ can be estimated by ball of radius $$(\varepsilon \tilde H_{\text{min}})^{-1}$$, knowing its second fundamental form $$\tilde h_{ij} \geq \varepsilon \tilde H_{\text{max}}\tilde g_{ij}$$ ? This is reasonable on an intuitive level, but how can I write down an explicit proof ?

Thanks