Live Activity
The Live Activity page lists the giveaways, reaction goals and polls that are running or recently ended in a server.
What it lists
Every giveaway, reaction goal and poll the bot is running in the server, and the ones that have ended, with the prize or question, the channel, the number of entries or votes and the end time. Each row links to the Discord message and to the web summary page.
What you can do from it
See which events are about to end without scrolling channels, open the participants or voters list, and jump to the message to end or reroll it with the commands. Creating events still happens in Discord with /giveaway and /poll, because they need a channel to post in.
Not the activity log
The Activity log (in the dashboard header) is a different thing: it records settings changes with who made them and the old and new values. Live Activity is about events in the server; the log is about changes to the bot.
A custom bot running in the server shows its own events here, since it shares the same database.