i am studying convexity, and stumbled upon the statement and example below. Am i right to understand that the function in the example is convex because maximizing the equation on the right hand size with regards to V would give the value / equation on the left?
Yes, this is what the implied claim is. The right hand side is evidently a linear function of $z$, for every $v$. Taking the maximum over $v$ yields a convex function.
The fact that this function is $\log(1+e^{-z})$ is not entirely obvious; in fact, if I had to verify the convexity of the latter I'd probably do it by taking the second derivative.