It is a place where you can view the new activity in the projects and tasks that you are following. You will also receive info about the latest projects there.
After you open the task with new activity, the blue dot disappears. That means you already read the activity in this task. When you go to Activity view again that task will not be displayed on the list until your teammates will make the new activity in the task. When you enter such a task from a different view, it also makes the blue dot disappear. You can also mark this task as read with blue “read” button on the task list. Tasks in Activity view are grouped by project. You can also use blue “read” button next to the header with a project name. It will mark all tasks in this project as read.