Vector(int s) :elem{new double[s]}, sz{s} { }
It is the constructor. It is required an int parameter for instantiating.
elem{new double[s]}, sz{s} { }
This part describes how to initiate the member variables. An array type of double named elem. emem has "s" length array. sz is set as "s".
double& operator[](int i) { return elem[i]; }
This part is in order for access by element index.
Vector v(1);
return v[0]; // <= return double reference
I will share an example. This is introduced in The C++ Programming Language (4th Edition) written by Bjarne Stroustrup
#include <iostream>
using namespace std;
class Vector{
    public:
        Vector(int s) :elem{ new double[s]}, sz{s} {}
        double& operator[](int i) { return elem[i]; }
        int size() { return sz; }
    private:
        double* elem;
        int sz;
};
double read_and_sum(int s)
{
    Vector v(s);
    for (int i=0; i!=v.size(); ++i)
        cin >> v[i];
    double sum = 0;
    for (int i=0; i!=v.size(); ++i)
        sum += v[i];
    return sum;
}
int main() {
    int sum = read_and_sum(3);
    cout << sum << endl;
}