Influence - the act or power of producing an effect without apparent exertion of force or direct exercise of command.
This is why we "seek" a better trigger. Let's look at it this way. Have you ever driven a car or truck with "manual" steering? It takes alot of influence from the driver to steer. Power steering on the other hand takes much less influence to do the same job. Thus, both will achieve the same task but the power steering is more "precise" and "predictable".
Same goes for triggers. The trigger is the direct link between the shooter and the gun. The easier it is to "influence" the trigger the more control the shooter has over the gun. Your not fighting "resistance".
Your gun will shoot much more accurately because the trigger finger has less work to perform and is not influencing the desired result nearly as much. Plus, the "break" is easily anticipated and now the shooter can have a "repeatable" "consistent" feel.
The gun is more fun to shoot and better accuracy is achieved.
I hope that answers your question.