What does "terms evaluated in related environments yield related values" means in the context of typing judgements?
The environments are related not identical. The context of this statement is logical relations, as described e.g. here. A logical relation is a type-indexed family of relations which follows the structure of the types (in a way I won't elaborate on). If for each type $\tau$, we assign a (binary) relation $R[\tau]$, then given $f:\tau_1\to\tau_2,x:\tau_1\vdash f\, x : \tau_2$ when $R[\tau_1\to\tau_2](f,f')$ and $R[\tau_1](x,x')$ we can ask if it is the case that $R[\tau_2](f\, x,f'\, x')$. This is what it is referring to in "terms evaluated in related environments yield related result".