Live Activities allow applications to post detailed notifications on your Lock Screen, beyond the normal text message notifications. Examples might include shopping apps where Live Activities can give you rich details of the status of your order delivery a sports app might give you real-time scores of your preferred team, or a flight tracker app that can show real-time flight details on the Lock Screen, without you having to open the app.
The control simply lets you decide which (if any) apps can use this system.
From a privacy standpoint, it's not really doing anything that you aren't already doing - the shopping app knows what you've ordered, the sports app knows your favorite team(s), etc. It's just giving them a way to provide real-time (or near-time) feedback without you having to open the app.