Attach files on Activities created before attachments were stored in the Files object

If a user has the permissions to add attachments to Tasks and Events, they must edit the task or event in order to display the 'Attach File' button in the Attachments related list. The 'Attach File' button does not display in the Attachments related list when only viewing a task or event. 

This is in contrast to other standard objects which allow attachments from the parent record without clicking to edit the record first.

To navigate to 'Attach File' in Salesforce Classic 

1. Go to any task or events records (either via home tab or via open activities/activity history related list of any record).
2. Click Edit.
3. Scroll down to Attachments related lists, then click Attach File.

To navigate to 'Add Files' and 'Upload Files' in Lightning Experience

1. Go to any task or events records. 
2. Click "Related" section.
3. Click Add Files then select "Upload Files" or the "Upload Files" itself.