In the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $325$, there is the comment "any additional constraint will increase the value of the maximin", but without any proof of that. Is there anyone could explain to me rigorously why this is true? In fact, this is the main argument of the proof of the theorem $4$ on the same page.
The maximin is just defined in the theorem $2$ in page $324$.
That statement isn't quite true; what they mean (as reflected in the consistent use of weak inequalities in the preceding derivations) is the weak version of the statement: An additional constraint cannot decrease the value of the maximin. This is true because eliminating some of the options in the minimisation cannot lead to a lower minimum.
For example, say you have two sets, $\{1,2,3\}$ and $\{2,3,4\}$. Their minimal values are $1$ and $2$, respectively, so the maximin is $2$. Now you eliminate some of the values; say, you eliminate $2$ in both sets. Now the minimal values are $1$ and $3$, respectively, so the maximin is $3$. It increased; it could have stayed the same; but it couldn't have decreased, since the minimum of a subset of a set can't be lower than the minimum of the set.