Why does competition among traders affect how much of the gains from trade are given to the countries involved in the trade?

As competition increases, traders must offer certain advantage to their clients, e. g. lower prices, credit sales, longer payment terms, etc., which end up benefiting their clients, and also traders will be willing to relinquish some of their gains to keep existing clients.

This is exactly the same thing that occurs in a given market when the number of suppliers increases, decreasing the equilibrium price and increasing consumer surplus.