They describe a leaderboard built from server members' shared gaming activity, while actual rollout remains unconfirmed.
When It Appears
“See {name}'s recent activity” can appear on a server view that ranks shared gaming activity and shows a timed competition when Discord displays the link for opening a specific member's recent gaming activity.
What It Means
When “See {name}'s recent activity” is shown, the current step is presenting the link for opening a specific member's recent gaming activity.
A user who saw “See {name}'s recent activity” in Discord is likely searching to confirm that it means the link for opening a specific member's recent gaming activity, where it appears, and which state it describes.