If you really want to dig into the specifics of Smart Triggers, our article on the topic can be found here. If all you need is a quick rundown, here are the basics: Smart Triggers are a mechanical modification that turns the variable pressure trigger on the standard controller into a keypress, like on a keyboard.
This modification takes the time between pulling the trigger and receiving an in-game response and turns it to as close to zero as you can get. For competitive shooters and high actions-per-minute titles, this is an advantage highly sought after by talented players who need to squeeze every position for its maximum amount of advantage.
For casual players, the reasons one might want Smart Triggers can vary, but usually comes down to a subjective satisfaction with the *click* functionality over the “spring-y” pull of a trigger.