Since you only want to access certain files and not the whole backups, one method is to use the Finder app to drill down into the Time Machine (TM) backup and copy the files that you need over to your computer.
To do so:
By default, TM stores its backups on the Data partition on a Time Capsule (TC). We start by opening this partition using the Finder app.
- Open a Finder window.
- For the TC:
- Under the Shared section, click on the TC, and then, click on Connect As…
- Enter the disk password, and then, click on Connect.
- The Data disk partition should appear in the Finder window.
- Double-click on the disk partition to open it. Note: The disk partition should also mount to the macOS Desktop.
- Any available backup .sparsebundles should now appear in the Finder window. Each of these sparsebundles represent a TM backup set for a specific Mac.
Select the sparsebundle for his old computer; right-click on it, and then, select Open. As a result three things will happen:
A "folder," titled: Backups.backupdb, should now appear in the Finder window.
The "Time Machine Backups" drive will mount on the macOS Desktop.
"Time Machine Backups" will appear under the Devices section in Finder.
Click on the down-arrow for Backups.backupdb, then click on the down-arrow for the backup set that will appear. In this example that set is called: Europa. All of the existing backups (by oldest-to-newest date) in the set should now appear.
Click on newest date backup set and there should be a standard listing of all the files and folders on his computer as when last backed up.
Click on the down-arrow for Users, and then, click on the down-arrow for the specific user that you want to restore files from.