What does it mean in C++ when you have a parameter like ::lower?

But what does the scope operator mean when it doesn't have a first argument?

It denotes the global namespace.

Consider a simple example. Let;s assume that you defined function swap. At the same time you included header <utility> that also contains a declaration of swap. and after it dirrective using namespace std; To distinguihs them you need to use qualified name.

#include <iostream>
#include <utility>

using namespace std;


void swap( int &x, int &y )
{
    std::cout << "Inside ::swap" << std::endl;
    int tmp = x;
    x = y + 10;
    y = tmp + 10;
}

int main() 
{
    int x = 1;
    int y = 2;

    std::swap( x, y );

    std::cout << "x = " << x << ", y = " << y << std::endl;

    x = 1;
    y = 2;

    ::swap( x, y );

    std::cout << "x = " << x << ", y = " << y << std::endl;


    return 0;
}

The output will be

x = 2, y = 1
Inside ::swap
x = 12, y = 11

In this example ::swap refers to your function swap defined in the global namespace while std::swap refers to standard function swap declared in name space std in header <utility>

If you will write unqualified name

swap( x, y );

when the compiler will not know which function to call: either standard function std::swap or your own function ::swap

If to exclude directive

using namepsace std;

and write

swap( x, y );

then the compiler will call your own function swap.

Now I will explain why in your example with standard algorithm std::transform there is used ::tolower. The problem is that in C++ there are two functions with name tolower. One of which is C standard function declared in header <cctype> and other is C++ function declared in header <locale>. The C++ Standard allows to place C standard functions in the global namespace. So using name ::tolower allows the compiler to select the C standard function instead of the C++ function.