The feature of the League that was most opposed by Americans was the feature that called for what is called collective security.
In a collective security system, a group of countries make an agreement that says they will protect one another. If a country attacks one of these countries that make the agreement, all the other countries are supposed to help fight off the attack.
Many Americans were not at all happy about this idea. They did not think it would be a good thing if America were forced to fight in a war because some country invaded some faraway country that was part of the League of Nations. They thought it would pull the US into wars that really didn't matter for US security.
This is the main reason why Americans opposed the League.