You will need the Profile Unarchive permission to perform this action.
There are multiple ways to unarchive a person profile.
From Their Profile
Navigate to their profile and click on "Actions > Unachive" in the upper right area.
A pop up will appear asking you for the reason this person is being unarchived. Complete the required fields, and click on "Unarchive" to complete the action. These answers will be added as a note to the person's profile.
You will need both the People Archive and Reporting permission to unarchive in bulk.
Navigate to the People Query Builder in the left-hand menu, via Reporting > Query Builder > People, and construct a query that returns the people you are interested in unarchiving.
Next, click on "Mass Change" from the "Actions" menu right above the results area.
In the pop up that appears, select "Unarchive" from the "perform an action on profile" drop down.
A pop up will appear asking you for the reason you are unarchiving these people. Enter the required information, and click on "Done." You will be taken back to the "Perform Mass Change" pop up.
Finally, click on the update button to perform the bulk unarchive. The unarchive reason will be added as a note on the profiles that were unarchived.