Why do min/max operations in van Emde Boas trees run in $\mathcal{O}(1)$? I have read in the book 


  "Introduction to Algorithms", 3ed by Cormen, Leiserson, Rivest, and
  Stein (Chapter 20)


that Emde Boas trees run min/max operations in $\mathcal{O}(1)$ time since the minimum and maximum element are stored as attributes in each tree. I don't understand this very well. Does this mean that the min and max values are already given in the input, because the minimum and maximum must be calculated somewhere?
No. They are not given as an input! They were stored and updated as an attribute of each tree(Beware that the concept of the "attribute" in real-world programming means a "variable" in a programming language. More details depend on the programming language that you have chosen up to now to develop the data structure). The same situation is taking place in the min[Max]-heap data structure. You can find the minimum [maximum] in a min[max]-heap with $O(1)$ operation by keeping update (in insertion time) the mentioned attribute.