How insert OF a vector into a set works?

Your problem might be that you don't know what a set is. A set is container class like vector, except that you cannot insert the same element twice. In your case s is a set of objects of type vector<int>. Hence, s.insert(v) inserts the vector v into the set s.

To understand how a set works you might want to read:
http://www.cplusplus.com/reference/set/set/[^]

One of the important things is that a set keeps its elements in sorted order. When you insert a new element, a binary search is used to find the place where the element has to be inserted.

See http://www.cplusplus.com/reference/set/set/insert/[^] to read more about the insert operation.

Hope that made it a little clearer.

[AMENDED]

Here is a line by line description of what happens:

do {
    vector<int> v; // declares a vector of ints

    // this loop copies the values of the myints array
    // into v (note that the call to next_permutation will rearrange
    // the values in myints at the end of each iteration)
    for (int i = 0; i < N; i++)
       v.push_back(myints[i]);

    // inserts the vector v into the set s; as a set keeps its elements
    // in an ordered sequence, the insert function has to compare the new
    // element (v) with the elements already stored in the set. This comparison
    // is performed by the compare function specified in the sets
    // template parameters, and as no such parameter is specified, the
    // operation< member function of vector<int> is used, which compares two
    // vectors according to their lexicographical order. (Perhaps this is
    // the part you didn't understand)
    s.insert(v);

// calls std::next_permutations, which rearranges the values in
// the myints array; once the original permutation (0, ...9) has been reached,
// next_permutation will return false and the loop ends
} while ( std::next_permutation(myints,myints+10) );

[2nd AMENDMENT]

// loop over all elements of set s and assign each one in turn to a variable c
// (note: the auto keyword lets c assume the type that is the element type of
// set s, which is vector<int>)
for(auto c: s){

  // so now we have in c one of the vector<int> elements stored in s; in
  // the inner loop, we walk over all elements of that vector<int> and each
  // is in turn assigned to variable i
  for (auto i: c)
      // we output that element i and a space
      cout << i << " ";

  // after the inner loop has output all elements of c, we terminate
  // the output line
  cout << endl;
}

I hope that makes it clear what the nested loops do.